This paper discusses the implementation of linear model predictive control techniques with open-loop unstable models. These models come from unstable ...
Linear model predictive control of unstable processes Kenneth
R. Muske
and James B. Rawlings”
Department of Chemical Engineering, 78712, USA
The University
of Texas at Austin, Austin, Texas
Received 10 February 1993; revised 25 May 1993
This paper discusses the implementation of linear model predictive control techniques with open-loop unstable models. These models come from unstable plants and stable plants with conditionally stable disturbance models. The regulator is based on a quadratic performance criterion subject to input and state constraints. It is shown to be nominally stabilizing for all positive definite penalties on the input, positive semi-definite penalties on the output, and feasible constraints. Robust stability of the unconstrained regulator is also discussed. The case of incomplete state measurement is addressed with state estimation. The state is estimated from output measurements using the standard linear observer formulation. It is shown that the interconnection of a stable observer and the constrained regulator guarantees a nominally stabilizing controller. Keywords: model predictive control; constrained control; open-loop unstable processes
Although not prominent in the chemical and petroleum industries, open-loop unstable processes do occur in chemical process control applications. The most common occurrence is in integrating, or conditionally stable processes, such as tank levels and accumulation systems. In addition, stable processes are modelled as unstable systems in much of the model predictive control literature. In the dynamic matrix control algorithm, for example, the dynamics of the standard disturbance model, d(k + 1) = d(k), are conditionally stable. Although this disturbance model is viewed as a convenience to ensure integral action in the controller and removal of steady-state offset, the fact is that the augmented model dynamics are unstable. Recycle, exothermic reaction and heat integrated systems can also result in unstable linear models about a nominal operating point. The local unstable response is due to the non-linear dynamic behaviour of the process. For example, a non-isothermal CSTR with an exothermic reaction can exhibit three steady-state temperatures’. A linearized model about the middle steady-state temperature is an unstable model that represents the dynamic temperature response locally about the unstable steady state. Over a narrow range of operation, this linear model can adequately represent the non-linear process dynamics. In order to handle these applications within the linear
model predictive control framework, a formulation that addresses unstable systems is required. The model predictive control formulations based on convolution models, which include the common industrial implementations such as model algorithmic control or IDCOM (Richalet et aL2), dynamic matrix control, DMC (Cutler and Ramaker3), linear dynamic matrix control, LDMC (Morshedi et aL4), quadratic dynamic matrix control,
QDMC (Garcia and Morshed?), and IDCOM-M (Grosdidier et aL6), are unable to exactly model unstable plants. This imposes a limitation on the performance that can be achieved by these controllers due to the structural error in the process model. The model predictive control formulations based on transfer function or state-space models, such as shell multivariable optimizing control, SMOC (Marquis and Broustail’), generalized predictive control, GPC (Clarke et ~1.~),and receding horizon tracking control, RHTC (Kwon and Byun’), are able to model unstable plants. However, the implementation of these algorithms on unstable systems is not discussed. All of these approaches must also be tuned for nominal stability. This paper discusses the implementation of a nominally stabilizing linear model predictive control formulation discussed by Muske and Rawlings” on unstable plants.
Receding horizon regulator formulation ‘To whom all correspondence should be addressed. E-mail: jbraw@ che.utexas.edu
0959-1524/93/02008s-12 0 1993 Butterworth-Heinemann
Ltd
The discrete dynamic system model is the standard state-
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Linear model predictive control: K. R. Muske and J. B. Ra wlings
space formulation in which Y(k) is the vector of outputs, u(k) is the vector of inputs, and x(k) is the vector of states at time k. x(k + 1) = Ax(k) + h(k),
k = 0, 1, 2, . . .
(1)
y(k) = WV
The models considered in this article are assumed to be stabilizable and detectable. This includes minimal statespace representations of all transfer function models. The model predictive controller is based on a receding horizon regulator that minimizes the following infinite horizon open-loop quadratic objective function at each time k.
+ {u(j) - u,}~ R{u(j) - us} + Au(j)‘SAu(j))
(2)
Qy is a symmetric positive semidefinite penalty matrix on the outputs, with Ys the steady-state output target and Y(j) computed from Equation (1). R is a symmetric positive definite penalty matrix on the inputs in which u(j) is the input vector at time j in the open-loop objective function and U,is the steady-state input. S is a symmetric positive semidefinite penalty matrix on the rate of change of the inputs in which Au(j) = u(j) - u(j - 1) is the change in the input vector at time j. The vector U contains the N future open-loop control moves as shown below. u(k) u(k + 1)
c
u=
1
u(k + &-
mF Q(k) = x(k + N)Tax(k + N) + Au(k + N)T k+N-l x
SAu(k + N) + c (x(j)‘Qx(j)
1
+
rerun-)
+ A~(j)~sAu(j))
The output penalty term in Equation (2) is replaced with a state penalty term in Equation (5) in which state penalty. Q = C’Q,C is the corresponding Additionally, any positive semidefinite state penalty matrix Q is acceptable in this formulation. The terminal state penalty matrix, Q, is determined from the soluton of the following discrete Lyapunov equation.
c =
2
AT'QA'
= Q + ,@?A
i=o
Input and output constraints of the following form are considered: UminGU~‘)Gu~~~, j=k,k+
l,...,k+N-
l(7)
j= k + j,,k + j, + 1:. . . , k +j, (8)
AUmin 1, through time k + j,, j, 2 j,. The value of j, is chosen such that feasibility of the output constraints up to time k + j, implies feasibility of these constraints on the infinite horizon. The value of j, is chosen such that the output constraints are feasible at time k. The constrained regulator will remove the output constraints at the beginning of the horizon up to time k + j, in order to obtain feasible constraints and a solution to the quadratic program at time k. Rawlings and Muske” show the existence of finite values for both j, and j,. Consistency of the constraints in Equations (7), (8) and (9) requires that they form a non-empty feasible region such that x, satisfies Equation (8) and U, satisfies Equations (7) and (9). For unstable plants, the input sequence U in the optimization problem is chosen so that the unstable modes of the model prediction are zero at time k + N. If the unstable modes are not zero, they evolve uncontrolled and do not converge. This yields an unbounded objective function for Q > 0. Therefore the following equality constraint is appended to the quadratic program: z”(k + N) = p”x(k + N) = 0
(10)
Linear model predictive control: K. R. Muske and J. B. Rawlings
This constraint is also used for unstable plants when Q 2 0. In this expression, z” are the unstable modes and VUis determined from the Jordan form of A partitioned into stable and unstable parts in which the unstable eigenvalues of A are contained in J,.
straints Equations (7), (8) and (9) for stable plants or Equations (7), (8), (9) and (10) for unstable plants at time k = 0 implies feasibility of the quadratic program at every time k > 0. Let u(i(k) denote the optimal open loop input at time j determined from the solution of the feasible quadratic program at time k and x(jjk) denote the open-loop optimal state at time j determined at time k. At time k, the receding horizon regulator will inject the first open-loop optimal input, u(k) = u(klk), into the plant. Therefore, the value of the state at time k + 1 is x(k + 1) = x(k + Ilk). Since u(jlk) = 0 for all j 2 k + N and the constraints are specified on an infinite horizon, feasibility at time k implies that the following input is feasible at time k + 1. Proof.
The terminal state penalty for unstable plants is determined from the stable modes in a manner similar to Equation (6).
(11)
Straightforward algebraic manipulation of the quadratic objective presented in Equation (5) and the constraints in Equations (7) through (10) results in the following quadratic program for U: m$’ O(k) = UTHU + 2UT(Gx(k) - Fu(k - 1)) (12)
u(k + 2)k) i
fj=
u(k + N-
i
0
(13)
Ilk)
1
Feasibility at time k = 0 then implies feasibility at every time k > 0 by induction.
subject to: DU < d,x(k) EU = e,x(k)
u(k + Ilk)
+ dz
Construction of the quadratic program matrices and vectors, H, G, F, D, E, d,, d2, and el, and implementation of the model predictive controller based on this constrained receding horizon regulator formulation are discussed by Muske and Rawlings”. The restrictions on the penalty matrices, R > 0, Q > 0, and S > 0, in this formulation ensure H > 0. The quadratic program is then a strictly convex programming problem which implies that the solution is unique and global”.
Nominal constrained stability This section presents nominal constrained stability proofs for the constrained receding horizon regulator presented in the previous section with perfect measurements of the state at each time k. These, and several of the subsequent theorems, are generalizations of the stability theorems presented by Rawlings and Muske” in which only positive definite state penalties, Q > 0, and no input velocity penalty matrix, S = 0, were considered. The discussion begins with a proof of the feasibility of the constraints at every time k > 0 provided they are feasible at time k = 0. The subsequent theorems then prove nominal constrained stability for a non-zero initial state. 1. For all x0 E R”, feasibility of the quadratic program with objective function Equation (5) and con-
Theorem 1. For stable A and N 2 1, x(k) = 0 is an asymptotically stable solution of the closed-loop receding horizon regulator with objective function Equation (5) R > 0, Q 3 0, S 2 0, and feasible constraints Equations (7), (8) and (9) for all x0 E R”. Proof. Feasibility of the output constraints at time k = 0 is guaranteed by the selection of j, in Equation
(8)*. This implies the existence of a feasible solution to the quadratic program at every time k > 0 from Lemma 1. Let x(jlk) and u(jlk) denote the open-loop optimal state and input at time j determined from the solution of the quadratic program at time k and CD(k)denote the value of the objective function in Equation (5) at time k. The objective function value is @(k) = x(k)T Qx(k) + x(k + 1Ik)TQx(k + Ilk) + . . .
+ u(k)TRu(k) + . . . + u(k + N - l(k)T x Ru(k + N - Ilk) + Au(k)TSAu(k) + . . . + Au(k + N/k)TSAu(k + N(k) Since the input u(k) is applied condition at time k + 1 is x(k @(k + 1) represents the objective k + 1 with the input U defined in following relationship holds:
at time k, the initial + 1) = x(k + Ilk). If function value at time Equation (13), then the
Lemma
*This is shown by Rawlings
and Muske”
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Linear model predictive control: K. R. Muske and J. B. Rawlings a(k)
= x(k)TQx(k)
+ s(k
+ u(k)TRu(k) + Au(kj%SAu(k)
+ 1)
(14)
Optimization at time k + 1 yields an objective function value that is no greater than s(k + 1) which implies @(k + 1) < s(k + 1) and: Q,(k) > x(k)TQx(k)
+ u(k)TRu(k) + Au(k)=SAu(k)
+ @(k + 1)
(15)
Since R > 0, Q 2 0, and S > 0, the sequence {a(k)} is non-increasing and bounded below by zero, therefore it converges. This requires that u(k) converge to zero since u(k)TRu(k) converges to zero from Equation (15) and R > 0. The convergence of u(k) to zero then implies that x(k) converges to zero*. For unstable plants, the equality constraint in Equation (10) is stipulated at every time k. This requires (A,B) stabilizable and N 2 r, in which r is the number of unstable modes of A. This also restricts the set of feasible initial conditions due to the input constraints in Equations (7) and (9). The system can be stabilized if x0 E X, in which X, denotes the set of x0 for which there exists an input sequence {u(O), u(llO), . . . , u(N - 110)) satisfying the constraints in Equations (7), (9) and (10). A system is defined to be constrained stabilizable if and only if x0 E X. If a system is not constrained stabilizable, then the input constraints are too restrictive to control the unstable modes. In this case, the constrained system cannot be stabilized by any regulator. Since constrained stabilizability is a .function of the plant, input constraints and initial state, there are design options available to stabilize the system. These include increasing the manipulated variable action, decreasing the operating range and decreasing the magnitude of disturbances entering the system. The proof of nominal stability for stable plants in Theorem 1 is based on showing that u(k) converges to zero which implies that x(k) converges to zero. For unstable plants, the convergence of u(k) to zero is not sufficient to show that the unstable modes converge to zero. In this case, the proof of nominal stability requires that the solution of the quadratic program, U, converges to zero. Lemma 2 shows that this implies x(k) converges to zero. Lemma 2. Convergence of the solution of the quadratic program with objective function Equation (5) and feasible constraints Equations (7), (8), (9) and (10) to zero implies x(k) converges to zero.
The convergence of the solution of the quadratic Proof. program to zero implies that u(k) converges to zero and, therefore, that the stable modes converge to zero. Feasibility of the equality constraint on the unstable modes in Equation (10) implies that the input sequence U zeros the model prediction of the unstable modes, z”(k + N - Ilk), at every time k. Since U converges to
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zero and the equality constraint remains feasible, the unstable modes converge to zero. This implies x(k) converges to zero*. Theorem 2. For unstable A with (A,@ stabilizable and N > r, in which r is the number of unstable modes of A,x(k) = 0 is an asymptotically stable solution of the closed-loop receding horizon regulator with objective function Equation (5), R > 0, Q > 0, S 2 0, and feasible constraints Equations (7), (8), (9) and (10) for all XoEXp Proof. Since the feasibility of the input constraints at time k = 0 is guaranteed by x0 E X, and feasibility of the
output constraints is guaranteed by the selection of jr+, Lemma 1 implies the existence of a feasible solution to the quadratic program at every time k > 0. The convergence of u(k) to zero and Q(k) to some non-negative value follows in the same manner as Theorem 1. Since a(k) converges, the quadratic terms x(k)TQx(k), u(k)=Ru(k), and Au(k)TSAu(k) converge to zero from Equation (15). This implies a(k) converges to the same value as CD(k) from Equation (14). The value of Q(k) is determined from the optimal U and the value of s(k) is determined from v as follows: O(k) = UTHU + 2UTGx(k)
- 2u(k)TSu(k -
1)
+ xWTQxW s(k)
= uTHU + 2i?Gx(k) +
- 2u(klk -
l)TSu(k -
1)
xWTQx(k)
Convergence
of x(k)TQx(k) and u(k) to zero implies UTHU + 2UTGx(k) converges to m,(k) and @HU + 2uTGx(k) converges to s(k). Since m(k) and s(k) converge to the same value and H is positive definite, u converges to the optimal solution U. Therefore u(j + Ilk) converges to u(jlk) from the definition of u in Equation (13) and the convergence of u(k) to zero implies the convergence of U to zero by induction. The convergence of x(k) to zero then follows from Lemma 2. Corollary 1. The sequence of objective function values in Equation (5) for the receding horizon regulator satisfying the conditions in Theorem 1 for stable plants or Theorem 2 for unstable plants converges to zero. Proof. Theorem 2 shows the convergence of U and x(k) to zero for unstable plants, therefore the sequence {@t(k)} converges to zero for unstable plants. The convergence of U to zero for stable plants can be shown in the same manner as in Theorem 2. The convergence of x(k) to zero from Theorem 1 then implies the sequence {m(k)} also converges to zero for stable plants.
*The proof of this is a straightforward consequence tThis is shown by Rawlings and Muske”
of stable A.
Linear model predictive control: K. R. Muske and J. B. Rawlings
Figure 1 Output responses for the system in Example 1
Because of the delay in the system, j, must be set to 2 at time k = 0 to obtain feasibility of the output constraint. When the minimum input constraint is - 0.5, the regulator is able to reject the disturbance with N = 3. When the minimum input constraint is increased to - 0.2, the constraints in Equations (7) and (10) are infeasible for N = 3. Increasing N from 3 to 6 makes these constraints feasible and the regulator is able to reject the disturbance. When the minimum input constraint is further increased to -0.0101, the system is no longer constrained stabilizable. In this case, the input constraint is too restrictive to control the unstable mode of the system excited by the state disturbance. The dashed line in Figure 1 shows the conditionally stable response of the system from attempting to control the unstable mode by saturating the input at the minimum constraint value.
Incomplete state measurement The regulator presented in the previous section assumes that exact measurements of the state are available at each time k. This is not normally the case in most applications and, therefore, an observer is required to reconstruct the state from output measurements. The standard linear observer is formulated using the output measurements as follows: cZ(klk) = a(klk -
1) + Lb(k)
?(k + Ilk) = Af(kJk)
Figure 2
Input responses for the system in Example 1
Example 1
The following first order SISO unstable model was identified from step test data on a fluidized catalytic cracking unit operating in partial combustion mode with a sampling interval of 0.5 min. The model represents the effect of one of the combustion air flowrates on the regenerator bed temperature about the desired nominal operating point of the regenerator. The input and output variables are deviations from this nominal operating point. The second state is used to model the delay in the system.
-
CzZ(k(k -
1))
(16)
+ h(k)
In this expression a(klk) is the estimated value of the state x(k) given output measurements up to time k and L is the observer gain. A discussion of the construction of optimal observers for stochastic systems is presented by &tr6m’3 and Bryson and Ho’~. When using an observer to estimate the state, the open-loop objective function for the receding horizon regulator is computed by replacing X(J) with a(jlk) in the @(k) calculation of Equation (5). m:@(k)
= 2(k + NJk)TGi(k
+ A’jk) + Au(k + NT k+N-I
x
SAu(k + N) + c (3(j1k)TQa(jlk)
(17)
j=k
/t=[l.;l
A],
R=[$
C=[1.5
O]
In this example, the output target is zero with a maximum output constraint of 1.5. At time k = 0, a state disturbance of [I OITenters the system. This results in a disturbance of magnitude 1.5 in the output. The following regulator tuning parameters are used:
Q=[
2.25 0 0
01’
R=l,
S=O
Figures 1 and 2 show the closed-loop response of the system for three values of the minimum input constraint.
+
u(jlTRu(/] + A&)‘SAu(i))
The following recursion is used to compute a(jlk) starting with Equation (16). This is the optimal prediction of the state for the standard linear stochastic model of the system.
In the case of perfect state measurement, feasibility of the output constraints in Equation (8) at every time k is guaranteed by the selection ofj, at time k = 0. The value ofj, can then be decremented at each successive execution until j, = 1 without loss of feasibility. When an
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observer is used to reconstruct the state, however, the value of j, may have to be increased at time k > 0 to retain feasibility. This will be the case when, for example, the actual state is outside of the feasible region of the output constraint, but the state estimate is inside of this region. This can induce control action that causes an output constraint violation at a later time. In this situation, the value ofj, has to be increased. However, a finite value of j, always exists that ensures feasibility of the output constraint. Theorems 3 and 4 in this section prove nominal constrained stability of the receding horizon regulator formulation with state estimation for a non-zero initial state and reconstruction error. The reconstruction error of the state estimate in Equation (16) is determined as follows: t(k + 1) = x(k + 1) - 2(k + Ilk) = (A - ALC)c(k) (18)
This error converges to zero from an arbitrary non-zero initial condition if and only if A - ALC is stable, i.e. all eigenvalues are inside the unit circle. In this case, the observer is said to be stable. A necessary condition for stability of the observer is (C,A) detectable. The discussion that follows will be restricted to stable observers and, for simplicity, the penalty and constraints on Au will be neglected. Let u*(x) be the solution of the quadratic program in Equation (12) for an initial state x with S = 0 and subject to the constraints in Equations (7), (8) and (10). With perfect state measurement, the closed-loop system evolves as follows: x(k + 1) = Ax(k) + Bu*(x(k))
= T(x(k))
Theorems 1 and 2 show that x(k + 1) = T(x(k)) is asymptotically stable. The evolution of the closed-loop system with state estimation is: x(k + 1) = Ax(k)
+ Bu*(a(k))
The state estimate can be expressed in terms of the true state and the reconstruction error as follows: i-(k) = x(k) + (I - LC)&k)
Therefore, the closed-loop system with state estimation is governed by: x(k + 1) = T(x(k))
+ P(k,x(k))
in which the perturbation P(k,x(k))
= B[u*(x(k)
(19)
k > 0
With these preliminaries, we state and outline the proof of closed-loop stability with state estimation. Theorem 3. For stable A and N 2 1, x(k) = 0 is an asymptotically stable solution of the closed-loop receding horizon regulator with objective function Equation (17), R > 0, Q 2 0, S = 0, feasible constraints Equations (7) and (8), and stable observer Equation (16) for all x(0) E R” and c(O) E R”. Outline of proof. The key step in the proof is the use of a theorem from LaSalle”. This theorem states that asymptotic stability of T implies strong stability under perturbations in Equation (19) if T is Lipschitz continuous. Global Lipschitz continuity of the solution of feasible quadratic programs with unique solutions is proved by Hage?. Feasibility of the output constraints in Equation (8) at every time k is guaranteed by the selection of j,. Feasibility and strict convexity of the objective function ensure that the solution of the quadratic program in Equation (12) is unique at every time k. This implies that the solution is a globally Lipschitz continuous function of the state estimate. Lipschitz continuity of T follows from Lipschitz continuity of u*(x). Strong stability under perturbations implies that x(k) converges to zero in Equation (19) since t(k) converges to zero for a stable observer. The remaining step in the proof is establishing that P(k,x(k)) in Equation (20) satisfies the boundedness condition in LaSalle’s theorem, which is straightforward. For unstable plants with state estimation, the set of initial conditions is further restricted such that x(k) E X, and .f(klk) E X, for all k B 0. The first restriction ensures constrained stabilizability of the plant for all k > 0. The second restriction ensures that a feasible solution to the quadratic program in Equation (12) exists for all k > 0. Therefore, an unstable plant with state estimation can be stabilized if (x(0)&0)) E W, in which W, denotes the set of (x(0),5(0)) satisfying these restrictions. This is a sufficient condition for constrained stability with state estimation since the first restriction is stronger than is necessary to ensure that the unstable modes of x(k) can be brought asymptotically to the origin with a feasible input sequence for every k > 0. The set of initial states in W, is a subset of X, that depends on the value of the initial reconstruction error. As the initial reconstruction error approaches zero, the set of initial states in W, approaches the set X,.
due to state estimation is: + (I -
US(k))
-
u*HWl (20)
Since the reconstruction error is unaffected by the regulator, Equation (18) can be iterated to yield the reconstruction error at any time k:
90
k(k) = (A - ALC)k&O),
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Theorem 4. For unstable A with (AJ?) stabilizable and N > r, in which r is the number of unstable modes of A, x(k) = 0 is an asymptotically stable solution of the closed-loop receding horizon regulator with objective function Equation (17), R > 0, Q > 0, S = 0, feasible constraints Equations (7), (8) and (lo), and stable observer Equation (16) for all (x(O),&(O))E W,.
Linear model predictive control: K. R. Muske and J. B. Rawlings
system. Figure 3 also shows the predicted output computed from the estimated state in Equation (16). Because of the delay in the system, the value of j, must be increased to 2 at time k = 0 to achieve feasibility of the output constraints. At time k = 2, the value ofj, must again be increased to 2 to retain feasibility due to the reconstruction error. As shown in this example, with a poor observer the performance of the combined observer/regulator will be poor. However, the combined system is stable.
Steady-state offset and integral action Steady-state offset in the state and output can result from constant or asymptotically constant disturbances with the combined observer/regulator discussed in the previous section. The steady-state offset due to these disturbances can be removed by augmenting the plant model with a disturbance model. The states of this augmented disturbance model are not stable and not controllable. However, they are observable and the estimate of these states from the output measurements can be used to remove their influence from the system. This constant disturbance regulator formulation is discussed by Kwakernaak and Sivan” and is implemented through the determination x, and U, as described by Muske and Rawlings”. The only restriction in this approach is that an x, and U, that exactly remove the disturbance from each output may not exist. In this case, the implementation in Reference 10 removes the disturbance in a least squares sense. Since the regulator is stabilizing for the original system without the disturbance model, this formulation results in a stable closed-loop system.
1 B
Figure 4
Input response for the system in Example 2
Outline ofproof. Feasibility of the input constraints for all k > 0 is guaranteed by the restriction on the initial state and reconstruction error. Feasibility of the output constraints is guaranteed by the selection ofj,. This and strict convexity ensure that the solution of the quadratic program is a globally Lipschitz continuous function of the state estimate. Strong stability under perturbations and convergence of x(k) to zero follow as in Theorem 3. Example 2
Consider the unstable plant presented in Example 1 with an output target of zero, maximum and minimum input constraints of 1.0 and - 1.0, and a maximum output constraint of 1.5. The observer in Equation (16) is used to reconstruct the state from the output measurements. The regulator tuning parameters and observer gain are: R=l,
S=O,
N=5,
L=
This observer gain results in closed-loop observer poles of -0.8 and 0. At time k = 0, a state disturbance of [l OIT enters the system with a zero estimated state. Figures 3 and 4 show the closed-loop response of the
Output and state disturbance models
The standard method to obtain integral action in model predictive control implementations for stable plants is to assume that the difference between the model prediction and the measured output at the current time is caused by a step output disturbance that remains constant in the future. This approach can be represented within the state-space framework by augmenting the system in Equation (1) with a vector of additional states, d, that represents the output step disturbance.
[I;:‘+:;I=[A0 :I[a:;] +[:Iu(k)
(21)
The state estimates are updated as follows: i(k + l/k) = A_f(klk - 1) + h(k) ci(k + Ilk) = y(k) - C_i?(kjk- 1)
in which (2is the estimate of the output step disturbance
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and L = [0 ZlT is the filter gain for the augmented system. This filter gain is optimal for the stochastic system with no measurement noise and state noise affecting only the augmented d states. This formulation is equivalent to the observer in SMOC presented by Marquis and Broustai17. It has the advantage of being easy to implement, but it cannot be used with unstable plants since the observer poles contain the plant poles, i.e. the observer is unstable. For unstable plants, a stable observer for the augmented system in Equation (21) can be constructed if and only if the augmented (C,A) matrix pair is detectable. Note that this is not implied by the detectability or observability of the original (C,A). If the augmented system is detectable, a stable analogue of the standard observer formulation can be constructed simply by designing a filter gain that is optimal for the stochastic system with no measurement noise and state noise affecting both the augmented and the original states of the system. However, the performance of this combined observer/regulator in rejecting step disturbances can be rather poor, as demonstrated in Example 3. A second method for obtaining integral action is to augment the system with a step disturbance that affects the state. This approach can be represented by augmenting the system in Equation (1) with a vector of additional states, d, in which Gd represents the state step disturbance.
(22)
Again, in this formulation, the necessary and sufficient condition for the existence of a stable filter is the detectability of the augmented system. The closed-loop performance of these formulations is directly related to how well the disturbance model represents the actual disturbances affecting the system. A suitable model of the dynamic structure of the disturbances entering the system is required in order to accurately reject that disturbance and get acceptable performance. This subject is discussed by Francis and Wonham’* for linear and weakly nonlinear multivariable systems and has become known as the internal model principle. Lee et &.I9 also discuss this issue for a state-space formulation using the step response model coefficients. Example 3
Consider the unstable plant of Example 1 with the augmented disturbance models presented in this section. In this example, the performance of the combined observer/ regulator in rejecting output and input step disturbances is demonstrated. Each disturbance is unity magnitude and enters the system at time k = 0. The output target is zero and there are no input or output constraints. The regulator tuning
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J. Proc. Cont. 1993, Vol3, No 2
Table 1
Observer filter gain matrices and poles for Example 3 Output step
11
State step
1.326
Filter gain
L=
0
L=
-0.99
Observer poles
Output responses for the input and output step disturbances with the output step observer in Example 3 Figure 5
parameters, which result in deadbeat control action, are as follows:
e = [2f5i], R=O,
S=O,
N=2
A stable analogue of the standard observer formulation for the output step disturbance model in Equation (21) and an input step disturbance observer based on the augmented model in Equation (22) are used in this example. The observer filter gains and poles are presented in Table 1.
The observer filter gain for the output step disturbance model is the optimal filter gain for the stochastic system in which there is no measurement noise and the variance of the noise affecting the augmented state is 100 times that affecting each of the original model states. This covariance matrix was chosen to obtain a stable observer that closely replicates the standard output step disturbance observer for stable plants. With one of the closedloop observer poles at 0.99, the integral action is very slow and, as shown in Figure 5, the closed-loop performance of this combined observer/regulator is poor. Modifying the observer filter gain by increasing the variance of the noise affecting the original model states in relation to the augmented state noise variance does not significantly shift the closed-loop observer poles. In order to obtain acceptable closed-loop observer performance with this disturbance model, other observer tuning techniques, such as pole placement, must be employed. The observer filter gain for the state step disturbance model is the optimal filter gain for the stochastic system
Linear model predictive control: K. R. Muske and J. B. Rawlings
+ Au~‘)‘SAu@)
(24)
in which Q is a symmetric positive semidefinite penalty matrix on the outputs, S is a symmetric positive definite penalty matrix on the rate of change of the inputs, and AU is the vector of N + 1 future changes in the input.
1
lAu(k ‘+ N)] Figure 6 Output responses for the input and output step disturbances with the input step observer in Example 3
in which the state step disturbance is an input step, there is no measurement noise and there is state noise affecting only the augmented state. This formulation results in a deadbeat observer. Since the observer and regulator are, deadbeat, both disturbances can be rejected in three time periods as shown in Figure 6. However, this observer tuning is very aggressive and may not be useful in practice. This example makes clear that an appropriate disturbance model form is necessary for acceptable closedloop performance. The choice of a stable analogue to the standard stochastic observer formulation for integral action in stable plants is shown to exhibit poor performance.
Velocity form of the state-space model
Another method to obtain integral action is to augment the plant model as follows:
[;r++l;)]
= [:A
:] E;)]
+ [C!fjAu(k)
(23)
in which the augmented states are the output of the system and the new states and input represent the change in the original states and input (Prett and Garcia’@). This representation is obtained by differencing the state-space model in Equation (1) and augmenting with the output of the system. Unlike the previous formulation in which the augmented system model is used only to estimate the model and disturbance states and is not used by the regulator, this formulation uses the augmented system model in both the regulator and observer. Therefore, this approach is restricted to systems in which the augmented plant model is both stabilizable and detectable. The constrained, receding horizon regulator is constructed from the following infinite horizon open-loop objective function:
At time k + N + 1, the change in the input, dub), is set to zero and kept at this value for allj 2 k + N + 1 in the open-loop objective function value calculation. In order to have a bounded open-loop objective function in Equation (24) for all Q, there can be no steadystate offset in any of the outputs. A sufficient condition for no steady-state offset is that there are at least as many inputs as outputs and that there is a solution to the linear system in Equations (3) and (4). In the following, we will consider square systems, i.e. the number of inputs equals the number of outputs, in which the linear system in Equations (3) and (4) is full rank. Lemma 3 shows that the stabilizability condition on the augmented system in Equation (23) then guarantees that the regulator can determine an open-loop steady-state input that reaches the output target exactly. This ensures that the infinite sum on the output tracking error in Equation (24) is bounded. Lemma 3. Stabilizability of the augmented system in Equation (23) and full rank of the linear system in Equations (3) and (4) imply that for all y, there exists a AU input sequence for the augmented system that results in a unique steady-state input, u,, and a unique steady-state state, xs, that bring the output of the original system in Equation (1) to y, at steady state. Proof. The full rank condition on the linear system in Equations (3) and (4) implies that for all y, there exists a unique x,, u, such that Equations (3) and (4) are satisfied. The existence of a AU input sequence is shown as follows. Stabilizability of the augmented system implies that the unstable modes of any initial state can be brought to the origin with r input moves in which r is the number of unstable modes of the augmented system. Implementation of the input sequence AU(J) = 0, j > r keeps the unstable modes at the origin and also brings the stable modes to the origin. Choose as the initial condition the steady-state vector [0 -yT]’ and let u, = C&i AU(J)and x, = I,??, Axe. The infinite sum on Ax0 is on the stable modes converging to the origin for j > r; therefore, this sum converges. This Au0 sequence takes the origin of the augmented system to [0 y:]’ at
J. Proc. Cont. 1993, Vol3, No 2
93
Linear model predictive control: K. R. Muske and J. B. Rawlings steady state and the origin of the corresponding original state-space model to x = x, and y = y, at steady state. An equivalent finite horizon open-loop objective function is constructed as follows in which the origin of the states is shifted to [0 yTIT. inr
CD(~) = X(k + N)T&Y(k
+ N) + Au(k + N)T
x SAu(k + N)
(25)
k+N-I
i=k
X(k) =
1 1 A$
The terminal state penalty, Q, is determined from the solution of Equation (11) using the augmented system in Equation (23). The equality constraint on the unstable modes in Equation (10) is also determined using the augmented system. Since this formulation adds unstable modes through the augmented states, the equality constraint is required even when the original A matrix is stable. Input and output constraints of the same form as given earlier are considered in this formulation. The following theorems show nominal constrained stability for a non-zero initial state for both perfect state measurement and state estimation. In both cases, the set of feasible initial conditions is restricted due to the input constraints in Equations (7) and (9). These are the same restrictions that apply to the initial conditions in Theorems 2 and 4 for unstable plants.
Theorem 5. For the augmented system in Equation (23) satisfying the conditions in Lemma 3 and N B r, in which r is the number of unstable modes of A plus the dimension of the augmented state vector z, X(k) = 0 is an asymptotically stable solution of the closed-loop receding horizon regulator based on the augmented system in Equation (23) with objective function Equation (25), Q 2 0, S > 0, feasible constraints Equations (7), (S), (9) and (lo), and perfect state measurement for all X(0) E X,.
Proof. The existence of a feasible solution to the quadratic program at every time k follows as in Theorem 2. The following inequality results from the same steps as in Theorem 1: Q(k) 2 z(k)TQz(k)
+ Au(k)TSAu(k)
+ @(k + 1)
Since Q 2 0 and S > 0, the sequence {Q(k)} is nonincreasing and bounded below by zero. Therefore the sequence converges and Au(k) converges to zero. A positive definite Hessian in this formulation is guaranteed by S > 0. This implies the convergence of X(k) to zero in the same manner as the convergence of x(k) in Theorem 2. The convergence of u(k) and x(k) of the original system is shown as follows. The convergence of AU to
94
zero follows in the same manner as the convergence of U in Theorem 2. This implies u(k) converges to u(k + N + 1Jk) = u(k - 1) + cI”=‘k”Auo’lk). From Lemma 3, u(k + N + Ilk) = U, for all k, which implies u(k) converges to u,. The convergence of x(k) then follows from the convergence of u(k) and Ax(k). For the case of incomplete state measurement, the assumption of a stable observer in Theorem 6 is equivalent to assuming that the augmented system in Equation (23) is detectable. For stable A, Lee et a1.19state that the augmented system is detectable. Note that the standard output step disturbance observer for stable plants is also not a stable observer in this formulation.
J. Proc. Cont. 1993, Vol3, No 2
Theorem 6. For the augmented system in Equation (23) satisfying the conditions in Lemma 3 and N 3 r, in which r is the number of unstable modes of A plus the dimension of the augmented state vector z, X(k) = 0 is an asymptotically stable solution of the closed-loop receding horizon regulator based on the augmented system in Equation (23) with objective function Equation (25), Q 2 0, S > 0, feasible constraints Equations (7), (8) and (lo), and stable observer Equation (16) for all (~(OMO))
c w,.
Outline of proof. Feasibility of the input constraints for all k > 0 is guaranteed by the restriction on the initial state and reconstruction error. Feasibility of the output constraints is guaranteed by the selection of j,. Asymptotic stability of the origin from Theorem 5 implies X(k) converges to zero following the same argument presented in Theorem 3. The convergence of x(k) and u(k) follow in the same manner as shown in Theorem 5.
Example 4 Consider the plant model of Example tion presented in Equation (23).
,4=[11$:{
;J,
B=
[;I,
1 with the formula-
C=[OO
l]
In this example, the performance of the combined receding horizon regulator/observer with objective function Equation (25) is demonstrated with output and input step disturbances of unity magnitude that enter the system at time k = 0. The input and output are unconstrained with the following tuning parameters that result in deadbeat control. 0 Q=OOO, [0
0
0
0
11
R=O,
S=O,
N=3
The optimal observer filter gain for the stochastic system in which there is no measurement noise and the variance of the noise affecting the augmented state is 100 times that affecting the first state is L = [0.038 0 l.OIT.
Linear model predictive control: K. R. Muske and J. B. Rawlings Table 2
State feedback gains and stability regions for A, = 1.01
Q
R
K
1
_
_
2 2 2
10
1 1 10
1.010 0.927 0.689 0.533
N
Figure 7 Output responses for the input and output step disturbances in Example 4
Robust stability The model predictive controller formulation presented in this article requires the equality constraint in Equation (10) to stabilize unstable plants. There is a natural concern that this formulation, which depends on zeroing the unstable modes of the system for stability, might be sensitive to modelling errors. The input sequence computed as the solution to the quadratic program in Equation (12) zeros the unstable modes of the state prediction at time k + N. Injection of this sequence into the plant in an open-loop fashion almost certainly would not zero the unstable modes at time k + N due to the inevitable modelling errors. However, this fact does not imply that this formulation lacks robustness. Since this issue does not involve the inequality constraints, robust stability for the system without input and state constraints is considered. A linear state feedback regulator can be constructed for the unconstrained receding horizon regulator by transforming the equality constraint on the unstable modes into an arbitrarily large terminal state penalty on these modes. The terminal constraint penalty matrix is shown below in which QUis the positive definite penalty matrix on the unstable modes:
0.01 -0.07 -0.31 -0.47
< < <
