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Abstract– …låû˝Ö‰bÍ$ªW }−„Âqlí ½æ û˝íñ™uT|ø_®|-pä−ªÖ |‰íqlj¶ øqlí½æJ LQR í½æ õ&, FT|qlªJ°v5?4?D#U4íÛ ° ©_ä−FÛí|‰¡bÉs_ ¥_bñD }à PI −„Âí¡bbñøš qlíWäôý FT|íj¶óçªW ÉœÈ: Ö‰bÍ$ ªW }−„ (4ùŸ| ÂThe design of MIMO PI controller is for-mulated as an LQR problem. The weighting matrices of the quadratic performance index are chosen so that tuning can be done for each input-output channel and for tradeoff between transient response and robustness with respect to modeling error. The number of tuning pa-rameters is the same as that of a decentralized PI controller. A design example is given to demonstrate the feasibility of the proposed ap-proach.
1
IntroductionThe PI (proportional plus integral) controller is probably the most commonly used controller in the industry. Arguably the PI controller is the simplest practical controller that provides integral action which is required in many pro-cess control applications for asymptotic track-ing of setpoint commands and rejection of con-stant load disturbances [3],[11],[8]. There is much research on the design (or tuning) of PI controllers for SISO systems but very little is
done on MIMO design. Proportional plus inte-gral state feedback design, in the LQR frame-work, is discussed in [1] and [2], however the state estimator included for output feedback implementation gives away the simplicity of PI control. So far almost all the MIMO PI con-trollers proposed have a decentralized struc-ture, although some design include static pre-compensation to achieve diagonal dominance at steady-state [13], [4], [5]. In general decen-tralized structure limits performance although, being simpler, it may have some advantage in real-time implementations, e. g., fewer tun-ing parameters and easier to make the design fault-tolerant [10], [7]..
We formulate the design of MIMO PI con-troller as an LQR problem. The weighting ma-trices of the quadratic performance index are chosen so that tuning can be done for each input-output channel and for tradeoff between transient response and robustness with respect to modeling error.There are two tuning pa-rameters for each input-output channel. For low order plants, the number of input equals the number of output, the PI controller imple-ments exactly the optimal state feedback. For high order plants, the design involves approxi-mations: either model reduction of the plant or approximation of the feedback gain matrix or both. The error in the approximation is taken into account in robustness consideration and tuning can be done accordingly. A design ex-ample is given to demonstrate the feasibility of the proposed approach.
2
Problem FormulationConsider the linear time-invariant multi-input multi-output plant
˙x = Ax + Bu (2.1)
y = Cx (2.2)
where A ∈ IRn×n, B ∈ IRn×m and C ∈ IRm×n. The plant has m inputs and m outputs and the input-output transfer matrix is P (s) = C(sI −A)−1
B.We make the following assump-tions throughout
(A1) The plant is controllable, observable and (asymptotically) stable, and
(A2) The plant has no transmission zero at s = 0, that is, P (0) = −CA−1
B is nonsin-gular.
It follows from (A2) that A is nonsingular and both B and C are full rank. We note also that (A.2) is equivalent to that
"
−A B
C 0
#
is non-singular. The problem studied is the follow-ing. Given the MIMO plant (2.1) and (2.2) and a fixed PI controller structure, how do we design the PI gain matrices so that the closed-loop system is stable and achieves some perfor-mance requirements? The block diagram of the feedback system is shown in Figure 1, where Kp
and Ki are respectively the proportional gain
matrix and the integral gain matrix. We note that in decentralized PI controllers these gain matrices are constrained to be diagonal; in our proposed design they are in general full matri-ces. -d - Kp+Ki s - P(s) -6 + – r e u y
Figure 1: Closed-loop system with PI con-troller
For design purpose we will assume that the command input is a vector of step functions, i.e., r(t) = ¯r1(t) and ¯r ∈ IRm.
If the closed-loop system is stable, then Ki is nonsingular and the system reaches
con-stant steady-state as t → ∞. In steady-state, y(∞) = ¯r, e(∞) = 0, u(∞) = −(CA−1
B)−1 ¯ r, v(∞) = −K−1 i (CA −1 B)−1 ¯ r and x(∞) = A−1 B(CA−1 B)−1 ¯
r. Define the deviation vari-ables ˜x = x − x(∞), ˜v = v − v(∞), ˜u = u− u(∞)and ˜y= y − ¯r and rewrite the state equations of the closed-loop system as
" ˙˜x ˙˜v # = Ao " ˜ x ˜ v # + Bou˜ (2.3) ˜ y = [C 0] " ˜ x ˜ v # (2.4) ˜ u = −KpCx˜+ Kiv˜ (2.5) where Ao = " A 0 −C 0 # Bo = " B 0 # (2.6) We note that e = −˜y. We can think of the de-sign problem as one of constrained state feed-back design or if we take ˜v as part of the out-put (in addition to ˜y), as one of output feed-back design. The design goal here is to obtain good dynamic response of the tracking error e while maintaining a certain degree of robust-ness. Before discussing the design approaches, we show first that a stabilizing design exist, that is, there is always a nontrivial (Ki 6= 0)
PI controller which stabilizes the closed-loop system.
Proposition 1 Under the assumptions (A.1) and (A.2), there is a PI controller with Ki
non-singular so that the closed-loop system shown in Figure 1 is stable.
3
Design ApproachWe discuss the determination of the gain ma-trices Kp and Ki by LQR design .
Con-sider the system defined in (2.3). Let G = diag[α2
1,· · · , α 2
m] > 0 and let the quadratic
per-formance be defined as J = Z ∞ 0 ([˜x(t)T v(t)˜ T]Q " ˜ x(t) ˜ v(t) #
+˜u(t)TP(0)TRP(0)˜u(t))dt where Q = " CTGC 0 0 I # and R = diag[β2 1, . . . , β 2 m] > 0 Since e = −˜y= −C ˜x, the
first two terms of the performance index is sim-ply the weighted sum of square tracking error and sum of square integrated error. The choice of the third term requires some explanation. Define ˆu(t) = P (0)˜u(t) then the third term in-side the integral becomes ˆu(t)TRˆu(t). If we
think of ˜u as the input to the plant P (s) with output ˜y, then ˆuis the input to the ’normalized plant’ P (0)−1
P(s) to produce the same output ˜
y. Since P (0)−1P
(s) is diagonal at s = 0, it is nearly decoupled at low frequencies. Hence, roughly speaking, weighting a component of ˆu has the effect of weighting the control input re-quired for the performance of the correspond-ing component of output ˜y. The performance index can be written as
J = Z ∞ 0 (Σ m i=1α 2 i|ei(t)|2+|˜vi(t)|2+βi2|ˆui(t)|2))dt
where the subscripts i indicate the ith compo-nent of the respective vector. The parameters αi and βi are to be selected for the trade-off
between the tracking error response and the control effort required in each channel. If the response of every channel is of the same impor-tance then G and R can be chosen as a multiple of identity matrix (to start with.) Roughly, in-creasing R and dein-creasing G improves robust-ness at the expense of deteriorating dynamic response. The LQR solution gives a state feed-back control law
˜
u(t) = −(K1x˜(t) + K2v˜(t)) (3.7)
where K1 ∈ IRm×n and K2 ∈ IRm×m. The
fol-lowing result shows that the LQR control law (3.7) always gives a stable closed-loop system. Proposition 2 {Ao, Bo} is controllable and
{Q1/2, A
o} is observable.
Comparing (2.5) and (3.7), if we assign Ki := −K2 and if the proportional gain
ma-trix is such that
KpC = K1 (3.8)
then the PI control law and the state feed-back law are identical. The equation (3.8) has a unique solution if m = n, the number of outputs equals the number of states. In this case, Kp = K1C−1. We note that there are
many processes which can be adequately repre-sented by low order models staisfying the above condition[10], including models for rapid ther-mal processing systems [12], [9].
If the plant has more states than outputs, that is, m < n, then the equation (3.8) have no solution in general. So the LQR control law can not be implemented as a PI controller. One way to approach this problem is the fol-lowing. Perform a balanced model reduction on the plant to obtain a reduced plant model with the number of states equals the number of outputs, and then determine the gain matrices Kp and Ki by the LQR design with reduced
model. of the reduced system. How
Suppose ∆(s) is the additive model reduc-tion error transfer matrix, then a condireduc-tion for robust stability is for all ω
σmax(Hur(jω)) <
1 σmax(∆(jω))
(3.9) If condition (3.9) is satisfied with some margin, then the design can be expected to perform well for the original model. In general increase R will decrease kHurk∞. Of course we have
to make sure that the reduced model (which is stable) has a nonsingular dc-gain. A sufficient condition for the balanced reduced model to have a nonsingular dc-gain is given as follows. Suppose σ1 ≥ · · · ≥ σm > σm+1 ≥ · · · ≥ σn
are the Hankel singular values of the plant and ¯
σ1 ≥ · · · ≥ ¯σm are singular values of P (0).
If ¯σm > Σnj=m+1σj, then the reduced model
obtained using balanced realization by keep m states has a nonsingular dc-gain.
Another way to determine the gain matri-ces is to set Ki = −K2 and Kp as the least
square solution of (3.8), that is, Kp = K1CT(CCT)
−1
where K = [K1 K2] is the gain matrix
Error is now introduced to the supposedly op-timal state feedback controller. Performance of this approximated design depends on the error K1(I − CT(CCT)
−1
C), small error en-sures good performance. A combination of the two approaches above is to do model reduction keeping enough states (> m) to ensure small reduction error and to determine Kp by least
square approximation.
4
A Design ExampleWe illustrate the proposed design approach by the following example.
ExampleConsider the input output 2-state stable transfer matrix describing a high-purity distillation column near certain operat-ing point, P(s) = " 87.8 1+194s − 87.8 1+194s + 1.4 1+15s 108.2 1+194s − 108.2 1+194s − 1.4 1+15s #
The design specifications are:(a) Each chan-nel should have a step response that settles to within 10% of the desired final value within 40 minutes. (b) The design should allow for a worst-case time delay of one minute on the control action and for ±20% uncertainty in the actuator gains.
We note that the plant is ill-conditioned with the condition number 140 at s = 0. Since the uncertainty and unmodeled dynamics oc-cur at the input, plant input relative uncer-tainty model is used. We will take as the worst plant model as
Pa(s) = P (s)(I+∆a) exp(−s) = P (s)(I+Li(s))
where ∆a = diag[0.2 0.2] and Li(s) =
exp(−s)(∆a + I) − I. The design is to
re-main stable for the worst plant and to satisfy the time response specification. Let C(s) = Kp + Ksi. A sufficient condition for roust
sta-bility is for all ω σmax(CP (I + CP )
−1
(jω)) < 1 σmax(Li(jω))
(4.10)
This condition is checked as we tune the design parameters. A minimal realization is
A= " −0.0052 0 0 −0.0667 # , B = " 1 −1 0 1 # C = " 0.4526 0.0933 0.5577 −0.0933 #
Since the channels are treated as of equal im-portance, G and R are chosen scalar multiples of identity matrix. Initially they are chosen equal.
Figure 2 shows the robustness condition (4.10) is satisfied. The one minute time delay practically limits the bandwidth of the closed-loop system to about 1 rad/min. Step re-sponses of the closed-loop system is shown in Figure 3 and Figure 4. The design satisfies the time response requirement. Note that the response is better for the case where the step commands are of opposite signs.
5
ConclusionsWe have described a MIMO PI controller de-sign method based on LQR formulation. The choice of tuning parameters allow tuning of in-dividual input-output channel and tradeoff be-tween dynamic response and robustness. The number of tuning parameters is exactly the same as that of a decentralized PI controller. Although only setpoint command are consid-ered in the design, the same formulation is also applicable to design for load disturbance rejec-tion.
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[13] C. Vlachos, D. Williams and J. B. Gomm, “A genetic approach to decentralized PI controller tuning for multivariable pro-cesses,” IEE Proc. Control Theory Appl., vol. 146, no.1, pp.58-64, 1999. 10−2 10−1 100 101 10−2 10−1 100 101 frequency in rad/min
robustness bound and closed−loop gain
Figure 2: Design for robustness
0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time in min
step respone channel 1
0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 1.2 1.4
step respone channel 2
time in min
Figure 3: Step responses
0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 time in min
step respone channel 1
0 10 20 30 40 50 60 70 80 90 100 −1 −0.8 −0.6 −0.4 −0.2 0
step respone channel 2
time in min
Figure 4: Step responses positive and negative commands
