CONDITIONS AND METHODS FOR OFFSET-FREE PERFORMANCE IN DISCRETE CONTROL SYSTEMS

Transcription

1 CONDITIONS AND METHODS FOR OFFSET-FREE PERFORMANCE IN DISCRETE CONTROL SYSTEMS By YUZHOU QIAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2012

2 2012 Yuzhou Qian 2

3 To everyone under the same sky 3

4 ACKNOWLEDGMENTS The author is sincerely thankful to Prof. Oscar D. Crisalle, Distinguished Teaching Scholar and Professor in the Chemical Engineering Department of University of Florida, for helpful suggestions from the early stage of this work and doctoral candidates M. Rafe Biswas and Shyam P. Mudiraj for their constructive feedback on the revisions of this thesis. 4

5 TABLE OF CONTENTS page ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Background State Feedback PID Controller State observer and estimator Model Predictive Controller Method Description of Contents LITERATURE REVIEW Integral Action in PI Controllers Challenges Offset Performance in MPC Control Systems Offset-free Performance Considering Inconstant Signals ANALYSIS AND DESIGN OF PI CONTROLLERS System without Disturbance Description of System without Disturbance Controllability Offset Performance of System Rank of the Integral Gain K System with a Constant Disturbance Model Introduction of the System with a Constant Disturbance Model Problem Description The Offset Performance of the System Example Example Example 2 :CSTR Plant System Modeling equations Linearization of the system Offset performance

6 4 ANALYSIS AND DESIGN OF THE OBSERVER Stability of Regulation Problem Example ANALYSIS AND DESIGN OF THE MPC CONTROLLER Offest-free MPC Design Conditions for Constant Reference The Offset in the MPC When Tracking Nonconstant Reference Zero-offset in Output with Constant Reference Cause of the Offset with Nonconstant Reference New Control Structure and Method Summary Method Method Example and Comparison Ramp Reference Oscillating Reference FUTURE WORK Tracking Nonconstant Reference by Modifying the Disturbance Model Future Work APPENDIX A RANK PRESERVING THEOREM B THE CAYLEY-HAMILTON THEOREM REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 3-1 The parameters of the CSTR

8 Figure LIST OF FIGURES page 1-1 A typical block diagram of a feedback controller A typical block diagram of a system with the observer A block diagram of MPC controller A block diagram of MPC controller The diagram of CSTR tank system Comparison of close loop response to an ramp signal Comparison of close loop response to an unstable signal

9 Abstract of a Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science CONDITIONS AND METHODS FOR OFFSET-FREE PERFORMANCE IN DISCRETE CONTROL SYSTEMS Chair: Oscar. D. Crisalle Major: Chemical Engineering By Yuzhou Qian August 2012 The offset-free controller drives controlled outputs to their desired target, and this thesis addresses the problem of offset-free controller when tracking the constant or inconstant references. Conditions that guarantee detectability of the augmented system model are discussed both in proportional integral derivative control (PID) framework and model predictive control (MPC) framework. Also two examples are presented to introduce a new MPC method to guarantee the best closed-loop offset performance for unstable reference. 9

10 CHAPTER 1 INTRODUCTION This chapter is intended to be a guide to the topics covered by this thesis. The objectives of this work include the analysis of the stability and detectability of a proportional-integral controller, verification of the possible unstable performance in certain MPC controllers, and finally introduce an MPC method to obtain the improved offset performance in tracking inconstant reference. Several general state-feedback controllers and the challenges associated with their design are described in subsection 1.1. An example for the plant and its detailed information is provided in subsection 1.2. and subsection 1.3, and the organization of this thesis is explained in the description of contents section State Feedback PID Controller 1.1 Background The PID control, which is the abbreviation of proportional integral derivative control, is a general control mechanism used in the industry. Usually PID controller calculates the difference between a measured process variable and a desired set point as the error. The design of a controller attempts to minimize the error by adjusting the controller parameters. The PID controller involves three terms: the proportional term, the integral term and the derivative term, and these are usually denoted as P, I, and D respectively. The integral term is incorporated to track error states to reach its target value, and obtain the the offset-free performance.. In the absence of knowledge of the process details, a well designed PID controller is the best controller. A PID controller is called a PI controller when the derivative control is switched off. PI controllers are fairly common in the chemical related industries since derivative action is sensitive to unpredictable and meaningless noise. The disadvantage of this approach is that an anti-windup algorithm is required for the integral term to prevent an unnecessary penalty. 10

11 Figure 1-1. A typical block diagram of a feedback controller State observer and estimator A state observer provides an estimate of the state of the plant by measuring the input and output with necessary knowledge of the system. The observer structure is typically involved with computer-aid mathematical problems. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer. However, in most practical cases, the physical state of the system cannot be determined by direct observation. Instead, the indirect methods are used. Figure 1-2. A typical block diagram of a system with the observer Model Predictive Controller Method Model Predictive Control (MPC) is an optimal control method based on online numerical optimization. The control inputs and plant outputs are predicted by using a estimated system model, which is optimized at regular intervals according to the cost 11

12 function and constraints. The MPC method originally is a computational technique used to improve control performance in applications in process industries. Since then, predictive control became the most widespread advanced control method in the industry. MPC controller can work in the large scale multivariable systems, and provide a systematic method of dealing with constraints on inputs and states with simple design and tuning. One disadvantage of MPC is that the augmentation also faces the wind-up problem when sudden change occurs in the reference. Also the existing MPC methods only consider the step or constant reference signals for the zero offset performance, while in practice it is often desirable to have a inconstant reference. Figure 1-3. A block diagram of MPC controller. 1.2 Description of Contents The thesis material has evolved at University of Florida over the last one and a half years. The thesis is divided into four parts to examine and eliminate the offset performance for PI and MPC control system with constant or inconstant reference signals. Part I provides an introduction to this thesis, including the brief description about PID control method, state observer and estimator and the MPC control method. 12

13 Part II (Chapter 2) is concerned with the literature review. The conditions of offsetfree performance in the system with PI controller and MPC controller with the integral actions obtained by previous researchers are provided and discussed. Several disturbance models used by the pioneers to eliminate the influence of persistent disturbances and plant model mismatch are presented and discussed. Part III (Chapter 3 through 5) addresses the analysis and design of PI and MPC controllers under requirement of zero offset performance for both constant and inconstant reference signal. The fundamental mathematics method is used to discuss the rank of the integral gain K 1 and the existence of integral term z ( ), and the disturbance models. Also simulations of the plant system with different parameters are used to check the theory discussed in this part. Part IV (Chapter 6) is concerned with conclusions and recommendations for future work. Part V addresses the appendix, the reference and the the biographical sketch of the thesis. The appendix is composed of the mathematical methods necessary for the thesis and the unit conversion table for the plant system example. 13

14 CHAPTER 2 LITERATURE REVIEW 2.1 Integral Action in PI Controllers Integral controls have long been used in chemical engineering industries since 1970s to achieve the long term offset-free performance when the process is subjected to the unmeasured constant disturbance. Ogata (1987) considered a discrete square system with state equation and output equation, x (k + 1) = Ax (k) + Bu (k) (2 1) y (k) = Cx (k) in which x (k) R n is the plant state, y (k) R m is output or measured variable, u (k) R m is the control or manipulated variable.the matrices A R n n,b R n m and C R m n. The pair (A, B) is assumed to be controllable and the pair (C, A) is assumed to be observable. The integrator state equation is defined as z (k) = z (k 1) + r (k) y (k) (2 2) where r is the reference and the controller design of the control vector u (k) given by u (k) = K 2 x (k) + K 1 z (k) (2 3) The augmented state vector is chosen as x (k) u (k) If the reference is constant or a step function, u ( )and x ( ) exist under the following conditions: 1. The pair (A, B) is controllable and 14

15 2. The matrix A I B C 0 is full rank. (2 4) Since u (k) is designed to hold the relationship with x (k), z (k) as u (k) = K 2 x (k) + K 1 z (k) (2 5) If the z ( ) exists, for a step input, the augmented state vector would approach the constant vector with values u ( ), x ( ) and z ( ) respectively. The following equation at the steady state is obtained z ( ) = z ( ) + r y ( ) (2 6) Therefore, there is no offset error in the output when the input is a step or constant. 2.2 Challenges When the control is well designed as u (k) = K 2 x (k) + K 1 z (k), the terms u ( )and x ( ) are constants. u ( ) = K 2 x ( ) + lim k K 1z (k) (2 7) which means lim k K 1z (k) is a constant vector. However, this doesn t assure the z ( ) exists as a constant vector. The key problem is whether the square matrix K 1 is full rank. From Ackermann s formula, K = I m H GH G n 1 H 1 φ (G ) (2 8) 15

16 where K = K 2 K 1 and φ (G HK ) is the characteristic equation of G HK, and G = H = A 0 CA I m I n C B Since G HK satisfies its own characteristic equation, φ (G HK ) = 0, from the Cayley- Hamilton theorem. Thus, the expression of K is obtained as K = I m H GH G n 1 H φ (G ) (2 9) K 2 K 1 = K + 0 I m A I n CA B CB 1 (2 10) hence from the two equations above, K 1 = Im H GH G n 1 H A I n CA B CB I m 1 φ (G ) + 0 I m (2 11) Objective is to prove that K 1 is full rank. A simulation with random inputs is performed using Matlab using A as a 3 3 random matrix, B as a 3 2 random matrix, C as a 2 3 random matrix. These three matrices satisfy the relationships from Equation 2 1 to Equation The code operates for ten million cycles, finally obtaining 130 examples among which atleast one of the absolute value of pole (K 1 ) < This result doesn t show strong conclusion whether the matrix K 1 is full rank or not. The complexity of φ (G ) 16

17 and the fact that K 1 is a sub matrix of a larger matrix clearly increases the difficulty to prove directly. Another indirect method is applied in this thesis in the Section Offset Performance in MPC Control Systems Most model predictive control methods obtain offset-free control performance by adding the integrating disturbance to the process model. The purpose of these additional disturbances is to eliminate the mismatch between real system and the model or unmeasured disturbances by the corresponding integral action. It has been proved to be effective for particular conditions. This strategy is similar to the integration of the error in PID contorllers, which also brings similar disadvantages. Since the error integration is independent of the controller, this method may lead to wind up in constrained system. Thus, anti-wind up mechanisms are required generally. The methods aim to avoid this problem by employing a disturbance estimator approach. Thereby, the state update equations used for the prediction are augmented by the reference and disturbance. An observer is used to estimate the disturbance states, and the MPC is designed to reject the estimated disturbance and track the reference, and solve wind up problem. Figure 2-1. A block diagram of MPC controller. 17

18 2.4 Offset-free Performance Considering Inconstant Signals The existing methods essentially consider constant disturbance and reference and hence remove offset at steady-state. For more general signals, such as ramps and sines, these methods will fail to remove offset. U. Maeder (2010) provides a generalization of the disturbance estimation approach to arbitrary unstable dynamics. The reference signal is generated by a autonomous dynamic system x r (k + 1) = A r x r (k + 1) r (k) = C r x r (k) where A r R nr nr and C r R ny nr, and the matrix A r may be unstable. The signal is generated by mode λ with order p, if there exists a linear system such that s (k) = C s x s (k), x s (k + 1) = J λ,p x s (k) k = 0, 1,... where J λ,p is a Jordan block matrix for λ with order p, J = λ λ λ Offset-free performance is achieved under the assumption that the observer is stable and the following decompositions exist, y = u = m i=1 m i=1 y λ i p i u λ i p i 18

19 where λ i is the ith eigenvalue of A with the longest Jordan chain of length p i. This crucial points of the method are the choice of disturbance model satisfying the internal model condition, and the addition of target trajectory conditions to the MPC problems. The disadvantages of the method include increase of computational cost of the trajectory optimization and very limited improvement of the offset performance for certain reference due to the fact that the decompositions above don t hold generally, which is discussed in Chapter 6. 19

20 CHAPTER 3 ANALYSIS AND DESIGN OF PI CONTROLLERS 3.1 System without Disturbance Description of System without Disturbance A discrete-time, linear, time invariant model is presented and assumed to be completely state controllable. x (k + 1) = Ax (k) + Bu (k) (3 1) y (k) = Cx (k) in which x (k) R n is the plant state, y (k) R m is output vector,u (k) R m is the control vector with A R n n,b R n m and C R m n. The pair (A, B) is assumed to be controllable and the pair (C, A) is assumed to be observable. The integrator state equation is defined as z (k) = z (k 1) + r (k) y (k) where the reference r (k) R m and the control is designed as u (k) = K 2 x (k) + K 1 z (k) (3 2) Thus the state and the integrator state at the time k + 1 are z (k + 1) = z (k) + r (k + 1) y (k + 1) = z (k) + r (k + 1) C Ax (k) + Bu (k) (3 3) = CAx (k) + z (k) CBu (k) + r (k + 1) x (k + 1) = Ax (k) + Bu (k) = Ax (k) + B K 2 x (k) + K 1 z (k) 20

21 The states equations above can also be written as x (k + 1) z (k + 1) = or in the following format A BK 2 BK 1 CA + CBK 2 I m CBK 1 x (k + 1) z (k + 1) = G x (k) z (k) x (k) z (k) + Hw (k) I m 0 I m r (k + 1) (3 4) r (3 5) where w (k) = K G = x (k) z (k) A 0 CA I m I n H = B C K = K 2 K 1 The dynamics of the system are determined by the eigenvalues of the state matrix, and therefore related to controllability of the pair (G, H) Controllability Consider the controllability matrix H GH G 2 H G n+m 1 H (3 6) The PBH test requires (G si, H) to be full row rank for all s, and (G si, H) = A 0 CA I m si m+n B CB 21

22 When s = 1, the matrix becomes A I B C 0 and full row rank as assumed. When s 1 rank = rank = rank A si n 0 B CA (1 s) I m CB A si n 0 B 0 I m 0 A si n B + m since the knowledge of the controllability of the pair (A, B), the result is m + n, which indicates the pair (G, H) = A 0 CA I m, I n C B is controllable as long as The pair (A, B) is controllable The matrix A I n B C 0 is full rank Offset Performance of System The state of the discrete system above can be written as x (k + 1) = Nx (k) + Mr 22

23 where N = A BK 2 BK 1 CA + CBK 2 I m CBK 1 M = 0 I m At the time k = 1, 2,..., n, the system yields x (k + 1) = Nx (k) + Mr k = 0 x (1) = Nx (0) + Mr k = 1 x (2) = Nx (1) + Mr = N Nx (0) + Mr + Mr = N 2 x (0) + NMr + Mr (3 7) k = 2 x (3) = Nx (2) + Mr = N 3 x (0) + N 2 Mr + NMr + Mr Any k x (k + 1) = N k x (0) + N k j 1 Mr All eigenvalues of N are designed to be inside the unit circle, so the 1st term of Equation 3 7 with k the 2nd term of Equation 3 7 with k lim k N k x (0) = 0 lim N k 1 Mr = (I N) 1 Mr k 23

24 the steady output with k y s = = = C 0 m x s C 0 m (I N) 1 Mr A BK 2 BK 1 C 0 m I CA + CBK 2 I m CBK I m thus C 0 m I A BK 2 BK 1 1 = V y s CA + CBK 2 I m CBK 1 where V (A, B, C, K 1, K 2 ) is a matrix function of (A, B, C, K 1, K 2 ). Then C 0 m = V y s A BK 2 I m BK 1 CA + CBK 2 CBK 1 = (V y s C ) (A BK 2 ) V (V y s C ) BK 1 Since B is an arbitrary matrix, and C is full row rank, thus V = C (3 8) y s = r Hence the system works well with zero-offset performance Rank of the Integral Gain K 1 Notice the design of the output u (k) = K 2 x (k) + K 1 z (k) 24

25 The current problem is to check whether a non-full-rank integral K 1 exists when the eigenvalues of the matrix A BK 2 BK 1 CA + CBK 2 I m CBK 1 can be arbitrarily set by matrices K 1 and K 2. The matrix can be written as where E F CE I CF E = A BK 2 F = BK 1 And the matrices K 1 and K 2 are designed so that 1 is not one eigenvalue of the matrix, thus the matrix I E F CE I CF = I E CE F CF is full rank. With row operations, then the matrix is also full rank which also means I E F C 0 C (I E ) 1 F is full rank. Thus, the matrix F should be full column rank, and thus the matrix BK 1 is also full column rank. The integral gain K 1 needs to be full column rank. 25

26 3.2 System with a Constant Disturbance Model Introduction of the System with a Constant Disturbance Model The plant model is augmented with a disturbance model in order to capture the a constant disturbance model. The dynamics, disturbance models and output equation are generated : x (k + 1) d (k + 1) y (k) = Ax (k) + Bu (k) + B d d (k) = d (k) = d = Cx (k) + C d d (k) (3 9) where d (k) cannot be directly measured general in the practice Problem Description From the integral state equation, z (k) = z (k 1) + r (k) y (k) = z (k 1) + r (k) Cx (k) C d d (k) using the controller structure u (k) = K 2 x (k) + K 1 z (k) one can obtain the expression for time k z (k + 1) = z (k) + r (k + 1) y (k + 1) = z (k) + r (k + 1) C Ax (k) + Bu (k) C d d (k) = CAx (k) + z (k) C d d (k) CBu (k) + r x (k + 1) = Ax (k) + Bu (k) + B d d (k) = Ax (k) + B K 2 x (k) + K 1 z (k) + B d d (k) 26

27 Then these two parameters are used to obtain the state equation x (k + 1) z (k + 1) = A BK 2 BK 1 CA + CBK 2 I m CBK 1 x (k) z (k) + 0 I m B d C d r d (3 10) The Offset Performance of the System From Equation 3 10, the state vector at time k + 1 can be written as x (k + 1) = Nx (k) + Mr where N = A BK 2 BK 1 CA + CBK 2 I m CBK 1 Mr = 0 I m B d C d r d r = r d At the time k = 1, 2,..., n, the system yields x (k + 1) = Nx (k) + Mr k = 0 x (1) = Nx (0) + Mr k = 1 x (2) = Nx (1) + Mr = N Nx (0) + Mr + Mr = N 2 x (0) + NMr + Mr k = 2 x (3) = Nx (2) + Mr = N 3 x (0) + N 2 Mr + NMr + Mr Any k x (k + 1) = N k x (0) + N k j 1 Mr 27

28 For brevity, we do not repeat the process in Subsection here, one of the results of the process is that y s = r which means the system also works under the zero-offset performance. 3.3 Example Example 1 A example here is used to show the offset-free performance with and A = B = C = B d = C d = 0.01 with the constant disturbance d = h h s =

29 The controllability matrix of the pair (A, B) is = = ctrb (A, B) B AB A n 1 B which is full row rank. G = = A 0 CA I H = = I n C B 29

30 The controllability matrix of the pair (G, H) then use the place command = = ctrb (G, H) G GH G n 1 H place(g, H, ) obtain thus K = K 1 = 80 K 2 = the matrix gives A BK 2 BK 1 CA + CBK 2 I m CBK 1 eig (K ) = Thus the offset-free performance is obtained under this design. 30

31 3.3.2 Example 2 :CSTR Plant System Modeling equations A continuous stirred-tank reactor as shown in Figure 3-1 is considered in this thesis. An irreversible, first-order reaction, A B occurs in the liquid phase in the plant, and the reactor temperature is regulated with external cooling with the knowledge that the level is not constant. Mass and energy balances lead to the following nonlinear state-space model: Molar balance equation: dc F ( dt = 0 (c 0 c) k πr 2 0 c exp E ) h RT (3 11) Energy balance equation: dt F dt = 0 (T 0 T ) πr 2 h Mass balance equation: H ( + k 0 c exp E ) 2U + (T c T ) (3 12) ρc p RT rρc p dh F dt = 0 F (3 13) πr 2 where h is the level of the tank, c are the molar concentration, T is the reactor temperature, F is the outlet flow rate, T c is the coolant liquid temperature. 31

32 Figure 3-1. The diagram of CSTR tank system. In the plant control system above, the controlled variables are the level of the tank, h, and the molar concentration c. The third state variable is the reactor temperature, T, while the manipulated variables are the outlet flow rate, F, and the coolant liquid temperature, Tc. Moreover, the inlet flow rate is considered to act as an unmeasured disturbance Linearization of the system The open-loop stable steady-state operating conditions are the following h s = 0.659m, c s = 0.877mol/L, T s = 324.5K, F s = 100L/min, T s c = 300K. (3 14) Using a sampling period time of 1 min, a linearized discrete state-space model is developed in terms of the derivative states, derivative inputs and derivative outputs. 32

33 x = c c s T T s h h s, u = T c T s c, d = F 0 F s F F s 0 y = c c s T T s h h s (3 15) Table 3-1. The parameters of the CSTR. Parameter F 0 T 0 C 0 r k 0 E/R U ρ C p Nominal Value 100 L/min 350K 1 mol/l m min 1 8, 750K W /m 2 K 1 kg/l 0.239J/g K H J/mol The linear state space equation for the CSTR system: and its linear output equation x (k + 1) = Ax (k) + Bu (k) + B d d (k) (3 16) y (k) = Cx (k) + Du (k) (3 17) in which 33

34 A = (3 18) B = C = , D = (3 19) where d is the unmeasured disturbance in the plant system.the entries of the matrix C vary in different examples in this thesis, and all of them are presented Offset performance The controllability matrix of the pair (A, B) is = = ctrb (A, B) B AB A n 1 B

35 which is full row rank. However, with the matrices G = = A 0 A I H = = I n C B

36 The controllability matrix of the pair (G, H) = ctrb(g, H) is not full row rank, which indicates that the system performance has offset. 36

37 CHAPTER 4 ANALYSIS AND DESIGN OF THE OBSERVER Consider the discrete-time time-invariant system x (k + 1) = f (x (k), u (k)) (4 1) y (k) = g (x (k)) in which x (k) R n is the plant state, y (k) R m is output vector and u (k) R m is the control vector. A LTI system model is applied x (k + 1) = Ax (k) + Bu (k) (4 2) y (k) = Cx (k) where A R n n, B R n m and C R m n, assumed that the pair (A, B) is controllable, and the pair (C, A) is observable. The model is augmented with a disturbance model to capture the mismatch between Equation 4 1 and Equation 4 2. A common constant disturbance model is used x (k + 1) = Ax (k) + Bu (k) + B d d (k) d (k + 1) = d (k) y (k) = Cx (k) + C d d (k) where d (k) R n d. 37

38 The state disturbance estimator is designed based on the augmented model as follows: ^x (k + 1) ^d (k + 1) = A + B d 0 I L x L d ^x (k) ^d (t) + B 0 ( ) y (k) + C ^x (k) + C d ^d (k) u (k) (4 3) in most papers, they would require the gain L x and L d chosen to make the estimator stable. The properties L x and L d is discussed in this chapter. 4.1 Stability of Regulation Problem The state estimation error vectors defined to study the stability of the regulation problem of the system, ~x (k) = x (k) ^x (k) ~d (k) = d (k) ^d (k) therefore, subtract the equation Equation 4 3 from the equation Equation 4 2, ~x (k + 1) ~d (k + 1) = = A B d 0 I A B d 0 I ~x (k) ~d (t) L x L d L x L d ( ) C ~x (k) + C d ~d (k) C C d ~x (k) ~d (k) (4 4) the performance of the offset completely depends on the design of L x L d 38

39 The transpose of the state matrix in equation Equation 4 4 shows as = A A B d 0 I B d 0 I = AT 0 B T d I T L x L d L x C T C T d L d C C d L T x T C C d The existance of the answer to the stability question is determinated by the controllability L T d T of the pair AT 0 B T d I, C T C T d A simple and direct method to consider is the PBH test again under two situations, AT si 0 C T B T d (1 s) I C T d s = 1 so that the matrix AT I C T B T d C T d need to be full row rank, which indicates that the transpose form is full column rank. A I C B d C d 39

40 s 1 so that the matrix rank = rank AT si 0 C T B T d (1 s) I C T d A T si C T + m is full row rank as long as the original system is observable. Thus, the zero offset may be achieved by the proper design of the pair. L x L d 4.2 Example The plant control system discussed in Section A = B = B d = C = eye (3) C d = zeros(3) 40

41 The controllability matrix is = = = C ( T A T C T ) A T n+n d 1 C T d B T (A d + I ) C T + C T d B T d C ( T A T C T ) A T n+n d 1 C T ( 0 B T (A d + I ) C T B T n+nd ) 1 d A i C T i=0 C ( T A T C T ) A T n+n d 1 C T ( 0 (A + I ) B T C T n+nd ) 1 d A i B T C T i=0 d With Something C T ( n+nd 1 i=0 A i ) C T + C T d which is full row rank. Thus an appropriate estimator can be designed for this system to obtain zero-offset performance. G = = AT 0 B T d I

42 H = C T = C T d then use the place command L = place(g, H, ) obtain the the estimator set L = which meets the requirement of offset-free performance with the pair L x = L d =

43 CHAPTER 5 ANALYSIS AND DESIGN OF THE MPC CONTROLLER Most methods involving MPC to achieve the zero-offset performance are comparable to the PID method of integration of the error. The tracking error is fed into a servo compensator of the controller, which contains a model of the disturbance and reference dynamics. The existing methods are designed to reject the constant disturbance and references. 5.1 Offest-free MPC Design Conditions for Constant Reference Consider a finite discrete-time, linear, time invariant system described as follows: x (k + 1) = Ax (k) + Bu (k) + B d d (k) + w k y (k) = Cx (k) + C d d (k) + v k (5 1) in which x (k) R n is the plant state, y (k) R m is output vector, and u (k) R m is the control vector, and A R n n, B R n m, C R m n, B d R n p, and C d R m p are matrices. Let the system be controlled by an observer-based controller with state feedback ^x (k + 1) ^d (k + 1) = A + B d 0 I L x L d ^x (k) ^d (k) + B 0 ( ) y (k) C ^x (k) C d ^d (k) ^u (k) (5 2) where the gains L x and L d are selected to ensure the stability of the estimator. Both estimated L x and L d and estimated B d and C d influence the poles, and thus affect the tracking performance. The relationship ^d (k + 1) = ^d ) (k) + L d (y (k) C ^x (k) C d ^d (k) is called output disturbance model. 43

44 The error vectors are defined as ~x (k) = x (k) ^x (k) ~d (k) = d (k) ^d (k) ~u (k) = u (k) ^u (k) When the equation Equation 5 2 is subtrated from the equation Equation 5 1, one obtains the error dynamics of the form ~x (k + 1) ~d (k + 1) = = = A BK CA + CBK A BK CA + CBK A 0 CA I m B d I m CB d B d I m CB d I n C ~x (k) ~d (k) L x L d BK B d L x L d ( ) C ~x (k) + C d ~d (k) C C d L x L d ~x (k) ~d (k) C C d ~x (k) ~d (k) tthe offset performance depends on the design of L x L d and BK B d For convenience, the following notation is introduced ~A = A B d, B ~ = B, C ~ = C C d 0 0 I, ~ L = L x L d 44

45 The control gain K is chosen as the solution of the following unconstrained infinite horizon quadratic optimization problem min = y T k Qy T k + ut k Ru k The estimator parameter ~ L is designed to keep the system stable, and the following formula ensures the non-bias for white noise disturbance ~L = ~ A ~ C T ( ~C ~ C T ) 1 where the predictor gain matrix is the symmetric positive semi-definite matrix solution of the discrete algebraic Riccati equation, = ~ A ~ A T ~ A ~ C T ( ~C ~ C T ) 1 ~C ~A T Since the estimator matrix A ~ L ~ C ~ is stable and using the conditions in Subsection and Section 4.1, the state and the disturbance estimates are to reach steady values for time k. Vectors x s and y s are used to denote steady-state values of the model state and output, respectively. The steady-state output error is defined as e s = y s Cx s C d d s from the equations Equation 5 2 at k, x s = Ax s + Bu s + B d d s + L x e s Notice that the existence of e s = e ( ) and d s = d ( ) 45

46 follows the discussion in Subsection and they must meet the same requirement in Subsection 3.1.2, in this case B d needs to be full rank. Then with the output equation at steady state, 0 = L d e s combining the equations, obtain x s x = (I A BK ) 1 L x e s in which (I A BK ) 1 exists since A + BK is a strictly stable matrix. Then the offset Hy s r = H (e s + Cx s + C d d s Cx s C d d s ) = H I + C (I A BK ) 1 L x es follows the discussion in Section 4.1 and it must meet the same requirement A I C So if the design of the controller satisfies the conditions, and also has B d C d null (L d ) null ( I + C (I A BK ) 1 L x ) there is zero-offset in the controlled variable. 5.2 The Offset in the MPC When Tracking Nonconstant Reference Zero-offset in Output with Constant Reference The existing methods are designed to reject the constant disturbance and references. For more general signals, such as ramps and sines, these methods will fail to 46

47 remove the offset. By defining x t (k) = x (k) y t (k) = y (k) + r where x t (k) and y t (k) are the absolute value of the state and output of the system. Now, the problem is turned from tracking problem to regulation problem. Consider the discrete-time time-invariant system for regulation problem, x (k + 1) = f (x (k), u (k)) (5 3) y (k) = g (x (k)) in which x (k) R n is the plant state, y (k) R m is output vector and u (k) R m is the control vector. A LTI system model is applied, x (k + 1) = Ax (k) + Bu (k) (5 4) y (k) = Cx (k) where A R n n, B R n m and C R m n, assume that the pair (A, B) is controllable, and the pair (C, A) is observable. The model is augmented with a disturbance model to capture the mismatch between Equation 5 3 and Equation 5 4. A common constant disturbance model is used x (k + 1) = Ax (k) + Bu (k) + B d d (k) (5 5) d (k + 1) = d (k) where d (k) R n d. 47

48 The state disturbance estimator is designed based on the augmented model as follows: ^x (k + 1) ^d (k + 1) = A + B d 0 I L x L d ^x (k) ^d (t) + B 0 ( ) y (k) + C ^x (k) + C d ^d (k) u (k) (5 6) y (k) = Cx (k) + C d ^d (k) in most papers, they would require the gains L x and L d chosen to maintain stability of the estimator. The error states are defined as ~x (k) = x (k) ^x (k) ~d (k) = d (k) ^d (k) ~u (k) = u (k) ^u (k) The difference between equation Equation 5 6 and equation Equation 5 5 is ~x (k + 1) ~d (k + 1) = = A BK B d 0 I A BK L xc L d C B d I ~x (k) ~d (k) ~x (k) ~d (k) L x L d C ~x (k) (5 7) 48

49 And the offset states y (k + 1) = Cx (k + 1) + C d ^d (k + 1) = C ~x (k + 1) + C ^x (k + 1) + C d ^d (k + 1) = C ~x (k + 1) + C C d A BK = = + C 0 A BK + L x C B d C C d L x L d L d C C C d A BK I C ~x (k) + B d 0 I CA CBK C d L d C CB d CA CBK CB d + C d B d 0 I ^x (k) ^d (k) ~x (k) ~d (k) ~x (k) ~d (k) ^x (k) ^d (k) + ^x (k) ^d (k) L x L d C ~x (k) (5 8) The offset-free performance is obtained when the system is tracking a constant reference and satisfies the conditions, A I C B d C d and null (L d ) null ( I + C (I A BK ) 1 L x ) 49

50 The terms lim ~x (k) = 0 k lim ~d (k) = 0 k lim (k) = 0 k ^x lim ^d (k) = 0 k Cause of the Offset with Nonconstant Reference If the controller is tracking a nonconstant reference, r = r (k) The MPC controllers for the tracking problems are applied separately for reference at each time, The discrete-time time-invariant system (0), x (k + 1) = f (x (k), u (k)) y (k) = g (x (k)) with reference r 0 (k) = 0 k 0 r 0 (k) = r (0) k > 0 The final steady states with zero-offset performance is obtained as y s0 and x s0. The discrete-time time-invariant system (1), x (k + 1) = f (x (k), u (k)) y (k) = g (x (k)) 50

51 with reference r 1 (k) = 0 k 1 r 1 (k) = r (1) r (0) k > 1 The final steady states with zero-offset performance is obtained as y s1 and x s1.... The discrete-time time-invariant system (i ), x (k + 1) = f (x (k), u (k)) y (k) = g (x (k)) with reference r i (k) = 0 r i (k) = r (i ) r (i 1) k i k > i The final steady states with zero-offset performance is obtained as y si and x si.... With constant references r = r (k), and the equilibrium conditions, y e (k) = k y si = r (k) i=0 x e (k) = x sk The definition of the states x (k) is the difference between the absolute value of the states and the local steady states x e (k) = x sk. In addition, the definition of the output states y (k) is the difference between the absolute value of the output states and the local steady states y e (k) = k i=0 y si = r (k). 51

52 Thus when tracking a nonconstant reference, the equation Equation 5 8 is changed to y (k + 1) = + CA CBK C d L d C CB d ~x (k) + ~x s(k+1) ~x sk ~d (k) + d ~ s(k+1) d ~ sk CA CBK CB d + C d ^x (k) + ^x s(k+1) ^x sk ^d (k) + ^d s(k+1) ^d sk +r (k + 1) r (k) (5 9) The difference between the equation Equation 5 8 and the equation Equation 5 9, the change in the final steady states, causes the failure to rejecting the output offset. The error would build up for some nonconstant reference, such as ramps with high slope New Control Structure and Method Summary Notice in the equation Equation 5 9. Since the output states at the steady states, y s = C ^x sk + C d ^d sk = 0 the steady states and steady disturbances are linear dependent. Moreover, since y sk = r (k) ^x sk is linear dependent on the r (k). Thus if the control u has the form like u (k) = K 1 x (k) K 2 r (k + 1) r (k) the output response is expected to have a significantly better offset performance. The following method is used to offset-free tracking under the nonconstant reference tracking. The Subsection and Subsection briefly summarize the main steps of the procedure proposed in this thesis. Two types of revised MPC methods are used. 52

53 Method 1 1. Compute the estimator gains L x, L d 2. Use MPC method (y (k) r (k)) T Q (y (k) r (k)) + u T (k) Ru (k) N 1 min k=0 + (y (N) r (N)) T P (y (N) r (N)) s.t.constraints hold. where the matrices Q, R, P, S are weight matrices to calculate the original control u 0 (k) = K 1 (x ). 3. Optimize the the gain K 2 under the constraints. 4. The revised control 1 is used u (k) = K 1 x (k) K 2 r (k + 1) r (k) Method 2 1. Computer the estimator gains L x, L d 2. The revised MPC problem is given by (y (k) r (k)) T Q (y (k) r (k)) + u T (k) Ru (k) N 1 min k=0 + (y (N) r (N)) T P (y (N) r (N)) + u T (k) S (r (N) r (N 1)) s.t.constraints hold. where the matrices Q, R, P, S are weight matrices. 3. The revised control 2 is obtained Ramp Reference 5.3 Example and Comparison For ramp reference, the difference in reference r (k + 1) r (k) = 53

54 where is a constant. From the equations Equation 5 9 and Equation 5 8, the tracking offset y (k + 1) = + CA CBK C d L d C CB d CA CBK CB d + C d +r (k + 1) r (k) C ( x s(k+1) x sk ) ~x (k) ~d (k) ^x (k) ^d (k) + C d ( ^d s(k+1) ^d sk ) Since the output vector are at steady states for regulation problem y s = C ^x sk + C d ^d sk = 0 the steady states and steady disturbances are linearly dependent. Thus the tracking offset for the ramp reference y (k + 1) = + CA CBK C d L d C CB d CA CBK CB d + C d ~x (k) ~d (k) ^x (k) ^d (k) +r (k + 1) r (k) If the control is designed as u (k) = K 1 x (k) K 2 r (k + 1) r (k) 54

55 the offset is modified to y (k + 1) = + CA CBK 1 C d L d C CB d CA CBK 1 CB d + C d ~x (k) ~d (k) ^x (k) ^d (k) + (I CBK 2 ) (r (k + 1) r (k)) which implies that the additional condition for the zero-offset performance for ramp reference is CBK 2 = I The convergence rate is the same as that in tracking a constant reference. Figure 5-1 is the simulation result for the MPC problem in Subsection (y (k) r (k)) T Q (y (k) r (k)) + u T (k) Ru (k) N 1 min k=0 + (y (N) r (N)) T P (y (N) r (N)) + u T (k) S (r (N) r (N 1)) s.t.constraints hold. where Q, R, P, S are weight matrices. 55

56 Figure 5-1. Comparison of close loop response to an ramp signal. The offset is reduced and the results show great advantage of the revised MPC method in tracking the nonconstant reference, especially for the ramp signal Oscillating Reference Another simple damped spring-mass system is studied to demonstrate the ability of the proposed control scheme to handle an unstable and oscillating reference: d dt x (t) = y (t) = 0 1 k/m 1 0 ρ x (t) x (t) + k/m 0 1 u (t) where k = 1, m = 1, and ρ = 0.1. The real plant however shall be slightly perturbed with k = 1.2 and ρ = The goal is to track an oscillating reference with zero offset. The reference model is r (k) = π π 0.1 k r (0) 56

57 Figure 5-2. Comparison of close loop response to an unstable signal. Figure 5-2 shows the closed loop response to the unstable reference. It can be observed that the proposed controllers MPC1 and MPC2 both behave with lower offset in tracking of the reference. The controller MPC 1 takes advantage of the term K 2 r (k + 1) r (k) that does not account in the control penalty. Thus it has a slightly better performance than the one of the controller MPC 2. 57

58 CHAPTER 6 FUTURE WORK 6.1 Tracking Nonconstant Reference by Modifying the Disturbance Model U. Maeder (2010) provides a generalization of the disturbance estimation approach to arbitrary unstable dynamics. The reference signal is generated by an an autonomous dynamic system x r (k + 1) = A r x r (k + 1) r (k) = C r x r (k) where A r R nr nr and C r R ny nr, and the matrix A r may be unstable. The signal is generated by mode λ with order p, if there exists a linear system such that s (k) = C s x s (k), x s (k + 1) = J λ,p x s (k) k = 0, 1,... where J λ,p is a Jordan block matrix for λ with order p, J = λ λ λ Offset-free performance is achieved under the assumption that the observer is stable and the following decompositions exist: y = u = m i=1 m i=1 y λ i p i u λ i p i 58

59 where λ i is the ith eigenvalue of A with the longest Jordan chain of length p i. This crucial points of the method are the choice of disturbance model satisfying the internal model condition, and the addition of target trajectory conditions to the MPC problems. However, the decompositions above don t hold for certain reference. Another problem is that some disturbance models would jeopardize the stability of the system. The estimator (L x, L d ) is designed to keep the system stable based on the conditions the matrix AT si 0 C T B T d (1 s) A d C T d is full row rank for all s. In the Chapter 3, the matrix A d = I so that when s 1, rank = rank AT si 0 C T B T d (1 s) I C T d A T si C T + m thus as long as the original system is observable, the zero offset may be achieved. In this case, if the matrix A d is not full rank, the condition becomes that the matrix AT si 0 C T B T d A d C T d must be full row rank, for {s : s is eigenvalue of the matrix Aand s 1 }. 6.2 Future Work The future object is to combine my method with U. Maeder s disturbance model. The brief summary of the main steps of the procedure proposed in this thesis: 1. Choose a reference model A r and C r 59

60 2. Choose a disturbance model A d, B d, C d. The dynamics matrix A d must incorporate an internal model of A r. It may contain additional dynamics of expected disturbances. The parameters B d and C d are chosen such that the augmented system is observable. 3. Computer the estimator gain L x, L d 4. Use MPC method (y (k) r (k)) T Q (y (k) r (k)) + u T (k) Ru (k) N 1 min k=0 + (y (N) r (N)) T P (y (N) r (N)) s.t.constraints hold. where Q, R, P, S are weight matrices to calculate the original control u 0 (k) = K 1 (x ). 5. Optimize the the gain K 2 under the constraints. 6. The revised control 1 is used u (k) = K 1 x (k) K 2 r (k + 1) r (k) or 1. Choose a reference model A r and C r 2. Choose a disturbance model A d, B d, C d. The dynamics matrix A d must incorporate an internal model of A r. It may contain additional dynamics of expected disturbances. The parameters B d and C d are chosen such that the augmented system is observable. 3. Computer the estimator gain L x, L d 4. The revised MPC problem is given by (y (k) r (k)) T Q (y (k) r (k)) + u T (k) Ru (k) N 1 min k=0 + (y (N) r (N)) T P (y (N) r (N)) + u T (k) S (r (N) r (N 1)) s.t.constraints hold. 5. The revised control 2 is obtained 60

61 APPENDIX A RANK PRESERVING THEOREM In control theory, the rank of a matrix can be used to determine whether a linear system is controllable or observable. A is assumed to be an m by n matrix over either the real numbers or the complex numbers. If B is any n k matrix with rank n, then rank (AB) = rank (A) (A 1) If C is any l m matrix with rank m, then rank (CA) = rank (A) (A 2) 61

62 APPENDIX B THE CAYLEY-HAMILTON THEOREM The Cayley Hamilton theorem, which is named after the mathematicians Arthur Cayley and William Hamiltom, as given below. Let the characteristic polynomial of A be defined as p (λ) = det (λi n A) where A is a given n n matrix, I n is the n n identity matrix, and "det" indicates the determinant operation. Since the entries of the matrix λi m A are polynomials in λ, the determinant is also a polynomial in λ. The Cayley Hamilton theorem states that if one substitutes the matrix A for λ in polynomial p (λ), one obtains the zero matrix: p (A) = 0 The powers of λ that become powers of A by the substitution should be computed by repeated matrix multiplication, and the constant term in p (λ) should be multiplied by the identity matrix, which can also be treated as the zeroth power of A, so that it can be added to the other terms. The theorem leaves to the conclusion that A n can be expressed as a linear combination of the lower matrix powers of A. Especially when the ring is a field, the Cayley Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. 62

63 REFERENCES 1 Gabriele Pannocchia and James B. Rawlings, Disturbance Models for Offset-Free Model-Predictive Control, AIChE Journal, Vol. 49, No. 2 2 P.Henrlk Wallman, James M. Sllva, and Alan S. Foss, Multivariable Integral Controls for the Fixed Bed Reactors, Ind. Eng. Chem. Fudam., Vol. 18, No. 4, Urban Maeder and Manfred Morari, Offset- free reference tracking with model predictive control, Automatica, 46 (2010) Gabriele Pannocchia, Robust disturbance modeling for model predictive control with application to multivariable ill-conditioned processes, Journal of Process Control 13 (2003) Urban Maeder and Manfred Morari, Offset-Free Reference Tracking for Predictive Controllers, Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec , Urban Maedera, Francesco Borrellib and Manfred Moraria, Linear offset-free Model Predictive Control, Automatica 45 (2009) B. Shafai, S. Beale, H.H. Niemann, and J. L. Stoustrup, LTR Design of Discrete- Time Proportional- Integral Observers, IEEE Transactions on Automatic Control, Vol. 41, No. 7, , Johan Akesson, Per Hagander, Intergal Action- A Disturbance Observer Approach, Proceedings of European Control Conference, Cambridge, UK, September Francesco Borrelli, Manfred Morari, Offset Free Model Predictive Control, Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, Dec , P. Lancaster, Jordan Chains for Lambda Matrices, II, AEQ. MATH, 292, Gabriele Pannocchia and Eric C. Kerrigan, Offset-free control of constrained linear discrete-time systems subject to persistent unmeasured disturbances, Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii USA, December Linh Vu and Daniel Liberzon, Supervisory Control of Uncertain Linear Time-Varying Systems, IEEE Transactions on Automatic Control, VOL. 56, NO. 1, JANUARY Katsuhiko Ogata, Discrete-Time Control System, Prentice-Hall International Editions,

64 14 E. F. Camacho and C. Bordons, Model Predicitive Control in the Process Industry, Springer, Dale E. Seborg, Thomas F. Edgar, Duncan A. Mellichamp, Process Dynamics and Control, Wiley, Second Edition 16 Katsuhiko Ogata, Modern Control Engineering, Prentice-Hall, Third Editions, Donald E. Kirk, Optimal Control Theory- An Introduction, Dover, Oscar D. Crisalle, ECH 6326 Introduction to Advanced Process Dynamics and Control, Coures notes, Oscar D. Crisalle, System Models for MPC, Course notes, Oscar D. Crisalle, Introduction to Optimal Control, Course notes, Geoffrey A. Williamson, Linear System Theory,Course notes, Mark Cannon, C21 Model Predictive Control, Course notes, 64

65 BIOGRAPHICAL SKETCH Yuzhou Qian received his M.S. Degree in Chemical Engineering from the University of Florida in the summer of 2012, Bachelor degrees in physics and in statistics form Peking ( Beijing) University in He is currently a student towards his Ph.D. degree at Department of Chemical Engineering at Rensselaer Polytechnic Institute (RPI). His research interests include control theory research and the latest applications of control methods entire research process, modeling, optimizing, controlling, and making impossible things possible. 65