- Email: [email protected]
Performance Enhancement and Robustness for Linear Systems With Saturating Actuators
Copyright @ IFAC Robust Control Design, Prague, Czech Republic, 2000
PERFORMANCE ENHANCEMENT AND ROBUSTNESS FOR LINEAR SYSTEMS WITH SATURATING ACTUATORS R. Reginatto .,1 A. R. Teel··,2 E. R. De Pieri·
• Department of A utomation and Systems Universidade Federal de Santa Catarina 88040-900, Florianopolis, SC, Brazil [email protected], [email protected] •• Center for Control Engineering and Computation University of California at Santa Barbara Santa Barbara CA 93106 teel@ece. ucsb. edu
Abstract: £2 stability for null-controllable linear systems with bounded controls is achieved by scheduling a parameterized state feedback control law constructed from a parameterized family of solutions of an Hoc-type Riccati equation. The scheduling is based on the magnitude of a virtual control signal and explicitly allows saturation and non-monotonicity (in time) of the scheduling parameter. These properties enhance convergence of the state by allowing the full utilization of the available control authority. Robustness to both a class of memory less input uncertainties and a class . of input-additive disturbances is also guaranteed. Simulation results for a case study illustrate the main features of the proposed control scheme. Copyright@ 2000 [FAC Keywords: Saturation control, scheduling algorithms, Riccati equations, robust stability
1. INTRODUCTION
1993; Teel, 1995b; Saberi et al., 1996) by synthesizing parameterized linear feedback control laws based on a parameterized family of solutions of Riccati-type equations. Semi-global stabilization with performance improvements is also considered in (Wredenhagen and Belanger, 1994; Saberi et al., 1996) .
Global stabilization oflinear systems with bounded controls requires, in general, non-linear control laws (Sontag and Sussmann, 1990; Teel, 1992; Sussmann et al., 1994; Teel, 1996; Megretski, 1996). As a matter of fact, in (Fuller, 1969; Sussmann and Yang, 1991) it is shown that global stabilization by means of linear control laws is impossible even for a chain of 3 or more integrators. Nonetheless, semi-global stabilization by linear feedback was proven in (Lin and Saberi,
Employing such parameterized control laws, global asymptotic stability and £2 stability properties have been achieved in (Teel, 1995a; Megretski, 1996; Suares, R. et all, 1997; Lin, 1998) . The approach is to schedule the parameterized control law according to the evolution of the state of the system. The state is confined to ellipsoidal positively invariant sets which are contained in the region of linearity of the closed-loop system.
I Author supported by CAPES. On leave from Electrical Eng. Dept. of Universidade Federal do Rio Grande do Sui, Av. Osvaldo Aranha, 103, 90035-190, Porto Alegre, RS, Brazil. 2 Research supported in part by NSF under grant number ECS-9896140.
427
An alternative scheduling algorithm is given in (Lauvdal, 1998) for a class of single-input systems.
Riccati equation (see (Teel, 1995a) for a complete statement) .
The effect of these scheduling algorithms can be understood as a "nonlinear controller gain adjustment" according to the state of the system. The underlining idea is to use low gain when the state is large and to allow the gain to be higher when the state is small. This idea sounds consistent with achieving good performance for the closed-loop system. However, due to conservative estimates, usually the assigned gain is smaller than it could be and consequently not all the available control effort is used.
Lemma 1. Let Q : R~o --+ Rnxn be a continuously differentiable matrix valued function such that lim{-+oc Q(~) = 0 and, for all ~ E R >o, Q(O > 0 and (aQ(~)/a~) < O. Then , for syst~m (1) there exist a finite, > 0 and a continuously differentiable matrix valued function P(O satisfying lim{-+oc P(O = 0 and, for all ~ E R~o, A' P(~)
a~
This paper focus on the scheduling of a parameterized control law obtained from an Hoc-type Riccati equation. The scheduling is based on the magnitude of a virtual control signal. The goal is to achieve less conservative results, and, consequently, performance improvements, by explicitly allowing saturation to occur up to certain levels and by allowing non-monotonicity of the scheduling parameter. Global asymptotic stability and £2 stability from an exogenous disturbance input to the state are achieved along with robustness with respect to a class of memoryless input uncertainties. By introducing a modification in the control scheme, robustness against a class of input-additive uncertainties is also achieved.
u = -B'P(~)x
(4)
(5)
is well known to semi-globally stabilize the origin of system (1) in the absence of disturbance (w == 0) (Teel, 1995b). Such control law has also been employed in (Teel, 1995a; Megretski, 1996; Suares, R. et all, 1997; Lin, 1998) where global stabilization and £2 properties were achieved. The approach employs a scheduling mechanism to vary ~ according to the evolution of the state of the system as measured by a ~-dependent quadratic Lyapunov function. The state of the closed-loop system is confined to ellipsoidal positively invariant sets which function as conservative estimates of the region of linearity. In this way, saturation is avoided and, in the absence of disturbance, the parameter ~ varies monotonically in time. In this paper, we present a scheduling algorithm for the parameterized control law (5) applied to system (1) which is based on a virtual control signal and does not rely on ellipsoidal invariant sets. Saturation is explicitly allowed and the scheduling parameter is allowed to be non-monotonic in time. £2 stability from w to x is achieved along with robustness with respect to static input nonlinearities . A modification on the control scheme is proposed to account for input-additive uncertainties.
2. PROBLEM STATEMENT Consider the linear system with input saturation
= Ax + BO'(u) + B 2 w
= 0 (3)
Using the matrix valued function P(O from Lemma 1, the parameterized control law
The paper is organized as follows. Section 2 states the problem, background, and notation. The main result is presented in section 3, which also discusses implementation issues. Robustness aspects are developed in section 4. Finally, in section 5, an illustrative example is presented and section 6 states concluding remarks.
:i;
+ P(~)A + P(~)DP(~) + Q(O P(O > 0, ap(~) < 0
(1)
where x E Rn is the state, u E R m is the control input, w E RP is an exogenous disturbance, and 0'( .) is the standard decentralized saturation function of magnitude 6, i.e.,
3. MAIN RESULT
(2)
where sat<.\(vi) = sign(vi) max{lvil, 6}. Throughout the paper we assume that the pairs (A, B) and (A, B 2 ) are stabilizable and that no eigenvalue of A has strictly positive real part. These are standard assumptions for global stabilization of linear systems with bounded controls (Sontag and Sussmann, 1990; Teel, 1995b).
We present the main result for system (1) without accounting for input-additive uncertainties. Robustness properties will be dealt with in section 4. To state the main result, we introduce the following definition. Definition 2. Let P(~) and Q(~) be as in Lemma 1, 6 be as in the definition of the saturation function 0' (.), and consider the following:
We employ the construction of parameterized controls laws based on the parameterized Hoc-type
428
1. Let
(3 , 1'\" P be real constants S .t. 0 E (0, 1), p :::; 2.6., (3 E [0, (2.6. - p) / p] and define
0,
o < I'\, <
M(O
:=
oQ(O
+ j3P(OBB' P(O
(6)
~(x) := min{~ E R >o, IB1P(Oxl :::; p} -
{
(7)
2. Let r : R >o x Rn ~ R>o be a locally Lipschitz function satisfying 'v'(~, x) E R~o x Rn\{o} (8)
3. Let W : R~o ~ [0,1] be a locally Lipschitz function satisfying W(s)
= {O, s 2:: 1,
s:::;
(9)
I'\,
1'\,/2
The next theorem presents the scheduling algorithm and the achieved properties for the closedloop system. The proof can be found in (Reginatto et al., 2000) and is here omitted due to space constraints.
Theorem 3. Let P(O and Q(O be as in Lemma 1. Let k > 0 be a constant, and consider the notation introduced in Definition 2. Assume W E £ ooe . Then, system (1) with the controller (5) and the scheduling algorithm
~
is given by (12) . It is clear that the minimization problem in (12) always has a solution since lim{-+oo P(O = O. Moreover, it guarantees that ~(t) is non-decreasing in time. In this phase, ~(t) is increased as necessary to keep v(x,O between the allowed limits as specified by p. On the other hand, whenever v(x(t),~(t)) < p , ~(t) will be non-increasing and, at least for v(x(t),~(t)) < 1'\" strictly decreasing towards the nominal value ~ = O. This recovery of ~(t) is dictated by the function r and ensures that ~(t) ~ 0 whenever x(t) ~ O.
Remark 6. Since v depends on ~(t), the scheduling algorithm is defined as an implicit relation in ~(t) . However, since W = 0 for v 2:: 1'\" ~(t) will be constant during any transition between (12) and (13). As a result, only the state is able to drive such transitions thus ensuring that the algorithm is well defined. We will refer to the signal vet) in (10) as the size of a virtual control signal. The reason should become clear in section 4. Notice that, the virtual control signal is allowed to exceed the saturation limit .6. as specified by the parameter p. In this way, saturation of the actual control signal u is explicitly allowed. Any p-norm, pE [1 , 00), can be used in (10) .
3.1 Implementation issues v(x(t), ~(t)) := IB' P(~(t))x(t)1 ~(t)
= 9(~(0),
x(t), v«x(t),~(t)))
(10)
The minimization problem (12) required in the scheduling algorithm consists in a nonlinear programming problem. However, since the decision variable is scalar, the problem is easily tractable with standard algorithms. For instance, a simple search combined with the bisection algorithm can be successfully applied.
(11)
where ~(O) = ~(x(O)) and the operator 9 is such that ~ (t) satisfies ~(t)
=
min (,
if v(x(t), ~(t))
2:: P(12)
(E (((t-), 00)
Additionally, the function r(~ , x) required in (13) can be given a closed form expression . For instance r(C x) can be computed as
v(x(t),O:S P
~
= -sat(k 0 r(~, x) W(v(x, ~)),
otherwise(13)
where ~(t-) := limT_H_ ~(T), is such that x E £2 whenever W E £2, and, for w = 0, the equilibrium x = 0 is exponentially stable and globally asymptotically stable. Moreover ~(t) is bounded and limHoo ~(t) = O.
r(~,x)=-
x'M(Ox ,~
x
8{
(14)
x
Alternatively, r(~ , x) can be chosen independently of x. For instance, let r : R~o ~ R>o be a locally Lipschitz and globally bounded function satisfying
Remark 4. The scheduling algorithm can also be applied to schedule the parameterized solution of the algebraic Riccati equation associated with the LQR problem (equation (3) with B2 = 0) for system (1) with no disturbance w. The same stability properties of Theorem 3 hold take away the £2 stability.
-reo 8~~0 :::; M(~), reo
'v'~ E R~o
(15)
Then, satisfying (15) will also satisfy (8) for all (Cx) E R>o x Rn . The function r in (15) can also be g~en a closed form and it is computationally less expensive than (14) . The price paid is that the recovery of ~ may get considerably slower .
Remark 5. The scheduling parameter ~(t) is obtained either from (12) or (13), according to the value of v(x(t).c(t)) . As long as v(x(t).t(t)) = o. 429
Since the control scheme requires the solution of an optimization problem, we might be interested in a discrete-time implementation for the scheduling algorithm. Such implementation is possible by choosing an appropriate sampling time. However, since we have to guarantee a certain intersampling behavior, such implementation will sacrifice the global property of Theorem 3. Instead, a semi-global property will be obtained and, the smaller the sampling time, the larger the domain of stability. A similar reasoning holds for the exogenous disturbance which has to be bounded in this case.
:i;
= Ax
+ Ba(u + g(t,x)) + B 2 w
(18)
where g(t,x) is assumed to satisfy
Jg(t,x)J :::; h(Jxl), 'v'(t,x) E R x Rn
(19)
and h(·) is a locally Lipschitz and known function. We then have the result.
Theorem 9. Assume g(t, x) satisfies (19) and w E Then, the result of Theorem 3 holds for system (18) with the scheduling algorithm (12)(13) and the control law ( cm '
u
= -( 1 + q(x, 0) BI P(O x
(20)
4. ROBUSTNESS where q(x, 0 satisfies
In this section we point out robustness properties of the proposed control scheme and also propose modifications in the control law to account for input-additive d isturbances.
5 [h(JxJW
q(x,~) ~ (1 - a) xlQ(Ox
(21)
Compared to (5) , the control law (20) contains the additional term q(x, ~)BI P(Ox which is responsible to dominate the effect of the disturbance g . Notice, however, that the scheduling algorithm is still guided by v , which is now different from the magnitude of u .
Input nonlinearities: Robustness against input nonlinearities is inherited from the optimality of the control law, i.e., for each constant ~, the control law (5) is optimal. Let 4>(u) := [4>dud,"',4>m(u m)) where 4>i('), i = 1,"' , rn , are locally Lipschitz nonlinear function satisfying (16)
5. CASE STUDY
Theorem 7. Let system (1) with u replaced by 4>(u) where 4>(,) is as defined above. Then, Theorem 3 remains valid.
As an illustrative example of the scheduling methodology, we will apply our design to a perturbed chain of 3 integrators given by
Theorem 7 is actually a consequence of the following result . with ~ = 5. The parameterized solution of the Hoo-type Riccati equation is constructed for 'Y ~ 4 and for the choice Q(E) = ~~diag{25E, 4E 2, E} where E .- 1/(~ + 1). It is given in closed form by 3
Theorem 8. Let a( ·) be any decentralized globally Lipschitz and globally bounded function satisfying for some ~ > 0 ula(u)
~ ulsat~(u),
'v'JuJ E [0,
2~]
(17)
Then , Theorem 3 is valid for any such a( ·).
PI P2 P3] P2 P4 P5 [ P3 P5 P6
Although we have considered the standard saturation function in the statement of the problem, Theorem 8 shows that the control scheme in fact admits more general saturation functions while preserving stability. This robustness property exploits the stability margin induced by the optimality property of the control law (5).
(23)
where
\I
= 5y'E, P6 = 1600E + llood, PI = P3P5 P5 = 0.5(p~ - E), P2 = P3P6, P4 = -P3 + P5P6
P3
Input additive-nonlinearities: The proposed scheme can be made robust against input-additive nonlinearities by employing the low-and-high gain control in a similar fashion as in (Lin, 1998) .
In this example, T(~, x) is chosen on-line to satisfy T(~, x) = _Xl M(Ox / Xl a~~o x, where a is chosen 0.9 and different values are of p are considered. In all simulations k = 10 and the scheduling
Consider that the system (1) is perturbed by an input-additive nonlinear and time-dependent term in the form
3
The solution actually solves the Heo Riccati inequality, i.e., (3) with ~ replacing = .
430
"r;:= ":-=-= .... ,
---
5
...
,
~
o
p = 10, a fact that is associated with the control signal having reached the allowed limits twice .
.
1
,
",
"' .
'":~''' 2
3
•
~~I s:---: ::I
: : : : : 1
5
6
7
8
9
10
t(sec)
o
Fig. l. Closed-loop system response for x(O) [8, 0, 0]' and p = 7.5 (solid); Linear response with ~ = 0 (dashed); Saturated system response without scheduling (dotted).
4
6
8
10
12
14
16
18
l
6
8
10
12
14
16
18
20
Qc 10'
o
2
2
4
4
6
8
8
10
12
14
16
18
20
12
14
16
18
10
12
1.
16
18
20
10
12
14
16
18
20
t(sec)
20 \ \ -:::- 10" \
;
..z.
1
0
-
20
H
1\
.,.-.1,'\
~ \
~-10
: : : : :1
10
8
Figure 3 compares the performance of the proposed scheduling algorithm with the mechanism proposed in (Megretski, 1996). For this comparison we used p = 10 in the scheme of (Megretski, 1996) so that saturation could occur. The same initial condition as in Figure 2 is used. The proposed scheduling algorithm produces a faster convergence of the state and allows all the available control effort to be used. Although saturation could occur in the scheme of (Megretski, 1996), it actually does not happen in this case, which is a consequence of the conservativeness of the scheme. The obtained response is similar to the one obtained with the proposed algorithm for p = 5, thus not allowing saturation.
: :-~:l
~":f : : : : ,;;2': : §J::---~---:
6
Fig. 3. Comparison of closed-loop system response (solid) and algorithm from (Megretski, 1996) (dashed) for x(O) = [300, 0, 0]' and p = 10.
Figure 1 shows the performance of the closed-loop system in response to a small initial condition on Xl . For comparison purposes, also the responses of the unsaturated closed-loop system and the saturated closed-loop system without scheduling (~ = 0, "It 2: 0) are presented. One can see that the scheduling algorithm yields an intermediate performance, and the control signal starts off saturated.
'~-'--:
4
t(sec)
algorithm is implemented in discrete time with a sampling period of 5ms.
2
2
"
I
\, \
-20
, ,
~~\....-L/~ ' >-;:-:r-~"'::""-,=---=:'-'==j \, ,..... , ... \
I
\ / 10
12
14
16
18
20
t(sec)
~':~"
20
t(sec)
Fig. 2. Closed-loop system response for x(O) [300, 0, 0]' and p = 10 (solid); p = 5 (dashed).
'
..
i
-5
""
"
-10
:
"."
"" " ".
"
""
"
L -~2-~4--6~---'8-~10--:'':-2-~'4:---':'6--:'':-8---=20 0
t(sec)
To illustrate the global property, Figure 2 shows the behavior of the closed-loop system for Xl (0) = 300, a large initial condition. The effect of the parameter p can be observed. For p = 10, the maximum amount of saturation is allowed and the state exhibits a faster convergence than for p 5 (no saturation). The behavior of the scheduled variable ~ (t) is also depicted in the figure, along with t:(t) (defined above). The non-monotonicity of the scheduling variable is apparent in the case
Fig. 4. Closed-loop system response (solid) with exogenous disturbance (dashed) for x(O) = 0 and p = 7.5. The behavior of the closed-loop system subject to an exogenous disturbance is illustrated in Figure 4. In this simulation, the disturbance is chosen wet) = 25 cos(27r /3 t) exp( -t/6) and the initial condition is zero. For this disturbance, the saturated closed-loop system without scheduling
=
431
goes unstable. The scheduling algorithm guarantees x E £2 as desired. Figure 5 shows the performance of the closed-loop system subject to input-additive uncertainties. A non linear inputadditive term g(t,x) = 2(X3)2 sin(xd is introduced in the system which becomes unstable for the initial condition x(O) = [8, 0, 0]'. Figure 5 shows the closed-loop system response with the robustification term 20 (X3)4 q(x,O = (1 _ 0) x'Q(Ox The result shows that the stability of the closedloop system is obtained and the performance is similar to the unperturbed case (compare with Figure 1).
-50L--~--~~--~--~5---6~~--~--~~10
t(sec) 10
5 °U·· 0; -5
:-:c-
+
":"-10
b
L-~--~--~--~~5--~--~~~-7--~10
t(sec)
Fig. 5. Closed-loop system response with inputadditive disturbance for x(O) = [8, 0, 0)' and p = 7.5. Dotted line: v(t).
6. CONCLUSION We presented a control scheme for linear systems with saturating actuators. A parameterized solution of an Hoo-type Riccati equation is scheduled according to the magnitude of a virtual control signal. The scheduling algorithm allows saturation up to a certain level and does not rely on confining the state inside ellipsoidal invariant sets. Computational requirements of the algorithm are not expensive. The control scheme guarantees £2 stability from a disturbance input to the state, and global asymptotic stability in the absence of disturbance. Moreover, the scheme is robust to a class of static input nonlinearities and an additional design is provided that guarantees robustness against a class of input-additive uncertainties.
7. REFERENCES Fuller, A.T. (1969). In-the-Iarge stability of relay and saturating control systems with linear controller. Int. Jornal of Control 10, 457-480.
Lauvdal, T . (1998). Stabilization of Linear Systems with Input Magnitude and Rate Saturations. PhD thesis. Norwegian University of Science and Technology. Norway. Lin, Z. (1998). Global control of linear systems with saturating actuators. A utomatica 34(7), 897-905. Lin, Z. and A. Saberi (1993). Semi-global exponential stabilization of linear sytems subject to input saturation via linear feed backs. Syst . Cont. Letters 21(3), 225-239. Megretski, A. (1996). £2 BIBO output feedback stabilization with saturated control. In: 13th Triennial IFAC World Congress. San Francisco, USA. pp. 435-440. Reginatto, R., A.R. Teel and E.R. De Pieri (2000). Stabilization of saturated linear systems: A scheduling mechanism based on the control signal. Technical report . Federal University of Santa Catarina. Florianopolis, Brazil. Saberi, A., Z. Lin and A.R. Teel (1996). Control of linear systems with saturating actuators . IEEE Trans. Aut. Cont. 41(3), 368-378. Sontag, E.D. and H.J . Sussmann (1990). Nonlinear output feedback design for linear systems with saturating actuators. In: 29th CDC. Honolulu, Hawaii. pp. 3414-3416. Suares, R. et all (1997) . £2 disturbance attenuation for linear systems with bounded controls: an ARE-based approach. In: Control of Uncertains Systems with Bounded Controls (S. Taubouriech & G . Garcia, Ed.) . Springer. Sussmann, H.J. and Y. Yang (1991). On the stabilization of multiple integrators by means of bounded feedback controls. In: 30th CDC. Brighton, England. pp. 70-72. Sussmann, H.J., E.D. Sontag and Y. Yang (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Trans . Aut. Cont. 39(12),2411-2424. Teel, A.R. (1992). Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst. Cont. Letters 18, 165-17l. Teel, A.R. (1995a). Linear systems with input nonlinearities: Global stabilization by scheduling a family of ll oo -type controllers. Int. J . Robust Nonlinear Control 5, 399-41l. Teel, A.R. (1995b). Semi-global stabilizability for linear null controllable systems with input nonlinearities. IEEE Trans . A ut. Cont. 40(1),96-100. Teel, A.R. (1996). A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Aut. Cont. 41(9), 12561270. Wredenhagen, G.F. and P.R. Belanger (1994). Piecewise-linear LQ control for systems with input constraints. A utomatica 30(3), 403416.
432
PERFORMANCE ENHANCEMENT AND ROBUSTNESS FOR LINEAR SYSTEMS WITH SATURATING ACTUATORS R. Reginatto .,1 A. R. Teel··,2 E. R. De Pieri·
• Department of A utomation and Systems Universidade Federal de Santa Catarina 88040-900, Florianopolis, SC, Brazil [email protected], [email protected] •• Center for Control Engineering and Computation University of California at Santa Barbara Santa Barbara CA 93106 teel@ece. ucsb. edu
Abstract: £2 stability for null-controllable linear systems with bounded controls is achieved by scheduling a parameterized state feedback control law constructed from a parameterized family of solutions of an Hoc-type Riccati equation. The scheduling is based on the magnitude of a virtual control signal and explicitly allows saturation and non-monotonicity (in time) of the scheduling parameter. These properties enhance convergence of the state by allowing the full utilization of the available control authority. Robustness to both a class of memory less input uncertainties and a class . of input-additive disturbances is also guaranteed. Simulation results for a case study illustrate the main features of the proposed control scheme. Copyright@ 2000 [FAC Keywords: Saturation control, scheduling algorithms, Riccati equations, robust stability
1. INTRODUCTION
1993; Teel, 1995b; Saberi et al., 1996) by synthesizing parameterized linear feedback control laws based on a parameterized family of solutions of Riccati-type equations. Semi-global stabilization with performance improvements is also considered in (Wredenhagen and Belanger, 1994; Saberi et al., 1996) .
Global stabilization oflinear systems with bounded controls requires, in general, non-linear control laws (Sontag and Sussmann, 1990; Teel, 1992; Sussmann et al., 1994; Teel, 1996; Megretski, 1996). As a matter of fact, in (Fuller, 1969; Sussmann and Yang, 1991) it is shown that global stabilization by means of linear control laws is impossible even for a chain of 3 or more integrators. Nonetheless, semi-global stabilization by linear feedback was proven in (Lin and Saberi,
Employing such parameterized control laws, global asymptotic stability and £2 stability properties have been achieved in (Teel, 1995a; Megretski, 1996; Suares, R. et all, 1997; Lin, 1998) . The approach is to schedule the parameterized control law according to the evolution of the state of the system. The state is confined to ellipsoidal positively invariant sets which are contained in the region of linearity of the closed-loop system.
I Author supported by CAPES. On leave from Electrical Eng. Dept. of Universidade Federal do Rio Grande do Sui, Av. Osvaldo Aranha, 103, 90035-190, Porto Alegre, RS, Brazil. 2 Research supported in part by NSF under grant number ECS-9896140.
427
An alternative scheduling algorithm is given in (Lauvdal, 1998) for a class of single-input systems.
Riccati equation (see (Teel, 1995a) for a complete statement) .
The effect of these scheduling algorithms can be understood as a "nonlinear controller gain adjustment" according to the state of the system. The underlining idea is to use low gain when the state is large and to allow the gain to be higher when the state is small. This idea sounds consistent with achieving good performance for the closed-loop system. However, due to conservative estimates, usually the assigned gain is smaller than it could be and consequently not all the available control effort is used.
Lemma 1. Let Q : R~o --+ Rnxn be a continuously differentiable matrix valued function such that lim{-+oc Q(~) = 0 and, for all ~ E R >o, Q(O > 0 and (aQ(~)/a~) < O. Then , for syst~m (1) there exist a finite, > 0 and a continuously differentiable matrix valued function P(O satisfying lim{-+oc P(O = 0 and, for all ~ E R~o, A' P(~)
a~
This paper focus on the scheduling of a parameterized control law obtained from an Hoc-type Riccati equation. The scheduling is based on the magnitude of a virtual control signal. The goal is to achieve less conservative results, and, consequently, performance improvements, by explicitly allowing saturation to occur up to certain levels and by allowing non-monotonicity of the scheduling parameter. Global asymptotic stability and £2 stability from an exogenous disturbance input to the state are achieved along with robustness with respect to a class of memoryless input uncertainties. By introducing a modification in the control scheme, robustness against a class of input-additive uncertainties is also achieved.
u = -B'P(~)x
(4)
(5)
is well known to semi-globally stabilize the origin of system (1) in the absence of disturbance (w == 0) (Teel, 1995b). Such control law has also been employed in (Teel, 1995a; Megretski, 1996; Suares, R. et all, 1997; Lin, 1998) where global stabilization and £2 properties were achieved. The approach employs a scheduling mechanism to vary ~ according to the evolution of the state of the system as measured by a ~-dependent quadratic Lyapunov function. The state of the closed-loop system is confined to ellipsoidal positively invariant sets which function as conservative estimates of the region of linearity. In this way, saturation is avoided and, in the absence of disturbance, the parameter ~ varies monotonically in time. In this paper, we present a scheduling algorithm for the parameterized control law (5) applied to system (1) which is based on a virtual control signal and does not rely on ellipsoidal invariant sets. Saturation is explicitly allowed and the scheduling parameter is allowed to be non-monotonic in time. £2 stability from w to x is achieved along with robustness with respect to static input nonlinearities . A modification on the control scheme is proposed to account for input-additive uncertainties.
2. PROBLEM STATEMENT Consider the linear system with input saturation
= Ax + BO'(u) + B 2 w
= 0 (3)
Using the matrix valued function P(O from Lemma 1, the parameterized control law
The paper is organized as follows. Section 2 states the problem, background, and notation. The main result is presented in section 3, which also discusses implementation issues. Robustness aspects are developed in section 4. Finally, in section 5, an illustrative example is presented and section 6 states concluding remarks.
:i;
+ P(~)A + P(~)DP(~) + Q(O P(O > 0, ap(~) < 0
(1)
where x E Rn is the state, u E R m is the control input, w E RP is an exogenous disturbance, and 0'( .) is the standard decentralized saturation function of magnitude 6, i.e.,
3. MAIN RESULT
(2)
where sat<.\(vi) = sign(vi) max{lvil, 6}. Throughout the paper we assume that the pairs (A, B) and (A, B 2 ) are stabilizable and that no eigenvalue of A has strictly positive real part. These are standard assumptions for global stabilization of linear systems with bounded controls (Sontag and Sussmann, 1990; Teel, 1995b).
We present the main result for system (1) without accounting for input-additive uncertainties. Robustness properties will be dealt with in section 4. To state the main result, we introduce the following definition. Definition 2. Let P(~) and Q(~) be as in Lemma 1, 6 be as in the definition of the saturation function 0' (.), and consider the following:
We employ the construction of parameterized controls laws based on the parameterized Hoc-type
428
1. Let
(3 , 1'\" P be real constants S .t. 0 E (0, 1), p :::; 2.6., (3 E [0, (2.6. - p) / p] and define
0,
o < I'\, <
M(O
:=
oQ(O
+ j3P(OBB' P(O
(6)
~(x) := min{~ E R >o, IB1P(Oxl :::; p} -
{
(7)
2. Let r : R >o x Rn ~ R>o be a locally Lipschitz function satisfying 'v'(~, x) E R~o x Rn\{o} (8)
3. Let W : R~o ~ [0,1] be a locally Lipschitz function satisfying W(s)
= {O, s 2:: 1,
s:::;
(9)
I'\,
1'\,/2
The next theorem presents the scheduling algorithm and the achieved properties for the closedloop system. The proof can be found in (Reginatto et al., 2000) and is here omitted due to space constraints.
Theorem 3. Let P(O and Q(O be as in Lemma 1. Let k > 0 be a constant, and consider the notation introduced in Definition 2. Assume W E £ ooe . Then, system (1) with the controller (5) and the scheduling algorithm
~
is given by (12) . It is clear that the minimization problem in (12) always has a solution since lim{-+oo P(O = O. Moreover, it guarantees that ~(t) is non-decreasing in time. In this phase, ~(t) is increased as necessary to keep v(x,O between the allowed limits as specified by p. On the other hand, whenever v(x(t),~(t)) < p , ~(t) will be non-increasing and, at least for v(x(t),~(t)) < 1'\" strictly decreasing towards the nominal value ~ = O. This recovery of ~(t) is dictated by the function r and ensures that ~(t) ~ 0 whenever x(t) ~ O.
Remark 6. Since v depends on ~(t), the scheduling algorithm is defined as an implicit relation in ~(t) . However, since W = 0 for v 2:: 1'\" ~(t) will be constant during any transition between (12) and (13). As a result, only the state is able to drive such transitions thus ensuring that the algorithm is well defined. We will refer to the signal vet) in (10) as the size of a virtual control signal. The reason should become clear in section 4. Notice that, the virtual control signal is allowed to exceed the saturation limit .6. as specified by the parameter p. In this way, saturation of the actual control signal u is explicitly allowed. Any p-norm, pE [1 , 00), can be used in (10) .
3.1 Implementation issues v(x(t), ~(t)) := IB' P(~(t))x(t)1 ~(t)
= 9(~(0),
x(t), v«x(t),~(t)))
(10)
The minimization problem (12) required in the scheduling algorithm consists in a nonlinear programming problem. However, since the decision variable is scalar, the problem is easily tractable with standard algorithms. For instance, a simple search combined with the bisection algorithm can be successfully applied.
(11)
where ~(O) = ~(x(O)) and the operator 9 is such that ~ (t) satisfies ~(t)
=
min (,
if v(x(t), ~(t))
2:: P(12)
(E (((t-), 00)
Additionally, the function r(~ , x) required in (13) can be given a closed form expression . For instance r(C x) can be computed as
v(x(t),O:S P
~
= -sat(k 0 r(~, x) W(v(x, ~)),
otherwise(13)
where ~(t-) := limT_H_ ~(T), is such that x E £2 whenever W E £2, and, for w = 0, the equilibrium x = 0 is exponentially stable and globally asymptotically stable. Moreover ~(t) is bounded and limHoo ~(t) = O.
r(~,x)=-
x'M(Ox ,~
x
8{
(14)
x
Alternatively, r(~ , x) can be chosen independently of x. For instance, let r : R~o ~ R>o be a locally Lipschitz and globally bounded function satisfying
Remark 4. The scheduling algorithm can also be applied to schedule the parameterized solution of the algebraic Riccati equation associated with the LQR problem (equation (3) with B2 = 0) for system (1) with no disturbance w. The same stability properties of Theorem 3 hold take away the £2 stability.
-reo 8~~0 :::; M(~), reo
'v'~ E R~o
(15)
Then, satisfying (15) will also satisfy (8) for all (Cx) E R>o x Rn . The function r in (15) can also be g~en a closed form and it is computationally less expensive than (14) . The price paid is that the recovery of ~ may get considerably slower .
Remark 5. The scheduling parameter ~(t) is obtained either from (12) or (13), according to the value of v(x(t).c(t)) . As long as v(x(t).t(t)) = o. 429
Since the control scheme requires the solution of an optimization problem, we might be interested in a discrete-time implementation for the scheduling algorithm. Such implementation is possible by choosing an appropriate sampling time. However, since we have to guarantee a certain intersampling behavior, such implementation will sacrifice the global property of Theorem 3. Instead, a semi-global property will be obtained and, the smaller the sampling time, the larger the domain of stability. A similar reasoning holds for the exogenous disturbance which has to be bounded in this case.
:i;
= Ax
+ Ba(u + g(t,x)) + B 2 w
(18)
where g(t,x) is assumed to satisfy
Jg(t,x)J :::; h(Jxl), 'v'(t,x) E R x Rn
(19)
and h(·) is a locally Lipschitz and known function. We then have the result.
Theorem 9. Assume g(t, x) satisfies (19) and w E Then, the result of Theorem 3 holds for system (18) with the scheduling algorithm (12)(13) and the control law ( cm '
u
= -( 1 + q(x, 0) BI P(O x
(20)
4. ROBUSTNESS where q(x, 0 satisfies
In this section we point out robustness properties of the proposed control scheme and also propose modifications in the control law to account for input-additive d isturbances.
5 [h(JxJW
q(x,~) ~ (1 - a) xlQ(Ox
(21)
Compared to (5) , the control law (20) contains the additional term q(x, ~)BI P(Ox which is responsible to dominate the effect of the disturbance g . Notice, however, that the scheduling algorithm is still guided by v , which is now different from the magnitude of u .
Input nonlinearities: Robustness against input nonlinearities is inherited from the optimality of the control law, i.e., for each constant ~, the control law (5) is optimal. Let 4>(u) := [4>dud,"',4>m(u m)) where 4>i('), i = 1,"' , rn , are locally Lipschitz nonlinear function satisfying (16)
5. CASE STUDY
Theorem 7. Let system (1) with u replaced by 4>(u) where 4>(,) is as defined above. Then, Theorem 3 remains valid.
As an illustrative example of the scheduling methodology, we will apply our design to a perturbed chain of 3 integrators given by
Theorem 7 is actually a consequence of the following result . with ~ = 5. The parameterized solution of the Hoo-type Riccati equation is constructed for 'Y ~ 4 and for the choice Q(E) = ~~diag{25E, 4E 2, E} where E .- 1/(~ + 1). It is given in closed form by 3
Theorem 8. Let a( ·) be any decentralized globally Lipschitz and globally bounded function satisfying for some ~ > 0 ula(u)
~ ulsat~(u),
'v'JuJ E [0,
2~]
(17)
Then , Theorem 3 is valid for any such a( ·).
PI P2 P3] P2 P4 P5 [ P3 P5 P6
Although we have considered the standard saturation function in the statement of the problem, Theorem 8 shows that the control scheme in fact admits more general saturation functions while preserving stability. This robustness property exploits the stability margin induced by the optimality property of the control law (5).
(23)
where
\I
= 5y'E, P6 = 1600E + llood, PI = P3P5 P5 = 0.5(p~ - E), P2 = P3P6, P4 = -P3 + P5P6
P3
Input additive-nonlinearities: The proposed scheme can be made robust against input-additive nonlinearities by employing the low-and-high gain control in a similar fashion as in (Lin, 1998) .
In this example, T(~, x) is chosen on-line to satisfy T(~, x) = _Xl M(Ox / Xl a~~o x, where a is chosen 0.9 and different values are of p are considered. In all simulations k = 10 and the scheduling
Consider that the system (1) is perturbed by an input-additive nonlinear and time-dependent term in the form
3
The solution actually solves the Heo Riccati inequality, i.e., (3) with ~ replacing = .
430
"r;:= ":-=-= .... ,
---
5
...
,
~
o
p = 10, a fact that is associated with the control signal having reached the allowed limits twice .
.
1
,
",
"' .
'":~''' 2
3
•
~~I s:---: ::I
: : : : : 1
5
6
7
8
9
10
t(sec)
o
Fig. l. Closed-loop system response for x(O) [8, 0, 0]' and p = 7.5 (solid); Linear response with ~ = 0 (dashed); Saturated system response without scheduling (dotted).
4
6
8
10
12
14
16
18
l
6
8
10
12
14
16
18
20
Qc 10'
o
2
2
4
4
6
8
8
10
12
14
16
18
20
12
14
16
18
10
12
1.
16
18
20
10
12
14
16
18
20
t(sec)
20 \ \ -:::- 10" \
;
..z.
1
0
-
20
H
1\
.,.-.1,'\
~ \
~-10
: : : : :1
10
8
Figure 3 compares the performance of the proposed scheduling algorithm with the mechanism proposed in (Megretski, 1996). For this comparison we used p = 10 in the scheme of (Megretski, 1996) so that saturation could occur. The same initial condition as in Figure 2 is used. The proposed scheduling algorithm produces a faster convergence of the state and allows all the available control effort to be used. Although saturation could occur in the scheme of (Megretski, 1996), it actually does not happen in this case, which is a consequence of the conservativeness of the scheme. The obtained response is similar to the one obtained with the proposed algorithm for p = 5, thus not allowing saturation.
: :-~:l
~":f : : : : ,;;2': : §J::---~---:
6
Fig. 3. Comparison of closed-loop system response (solid) and algorithm from (Megretski, 1996) (dashed) for x(O) = [300, 0, 0]' and p = 10.
Figure 1 shows the performance of the closed-loop system in response to a small initial condition on Xl . For comparison purposes, also the responses of the unsaturated closed-loop system and the saturated closed-loop system without scheduling (~ = 0, "It 2: 0) are presented. One can see that the scheduling algorithm yields an intermediate performance, and the control signal starts off saturated.
'~-'--:
4
t(sec)
algorithm is implemented in discrete time with a sampling period of 5ms.
2
2
"
I
\, \
-20
, ,
~~\....-L/~ ' >-;:-:r-~"'::""-,=---=:'-'==j \, ,..... , ... \
I
\ / 10
12
14
16
18
20
t(sec)
~':~"
20
t(sec)
Fig. 2. Closed-loop system response for x(O) [300, 0, 0]' and p = 10 (solid); p = 5 (dashed).
'
..
i
-5
""
"
-10
:
"."
"" " ".
"
""
"
L -~2-~4--6~---'8-~10--:'':-2-~'4:---':'6--:'':-8---=20 0
t(sec)
To illustrate the global property, Figure 2 shows the behavior of the closed-loop system for Xl (0) = 300, a large initial condition. The effect of the parameter p can be observed. For p = 10, the maximum amount of saturation is allowed and the state exhibits a faster convergence than for p 5 (no saturation). The behavior of the scheduled variable ~ (t) is also depicted in the figure, along with t:(t) (defined above). The non-monotonicity of the scheduling variable is apparent in the case
Fig. 4. Closed-loop system response (solid) with exogenous disturbance (dashed) for x(O) = 0 and p = 7.5. The behavior of the closed-loop system subject to an exogenous disturbance is illustrated in Figure 4. In this simulation, the disturbance is chosen wet) = 25 cos(27r /3 t) exp( -t/6) and the initial condition is zero. For this disturbance, the saturated closed-loop system without scheduling
=
431
goes unstable. The scheduling algorithm guarantees x E £2 as desired. Figure 5 shows the performance of the closed-loop system subject to input-additive uncertainties. A non linear inputadditive term g(t,x) = 2(X3)2 sin(xd is introduced in the system which becomes unstable for the initial condition x(O) = [8, 0, 0]'. Figure 5 shows the closed-loop system response with the robustification term 20 (X3)4 q(x,O = (1 _ 0) x'Q(Ox The result shows that the stability of the closedloop system is obtained and the performance is similar to the unperturbed case (compare with Figure 1).
-50L--~--~~--~--~5---6~~--~--~~10
t(sec) 10
5 °U·· 0; -5
:-:c-
+
":"-10
b
L-~--~--~--~~5--~--~~~-7--~10
t(sec)
Fig. 5. Closed-loop system response with inputadditive disturbance for x(O) = [8, 0, 0)' and p = 7.5. Dotted line: v(t).
6. CONCLUSION We presented a control scheme for linear systems with saturating actuators. A parameterized solution of an Hoo-type Riccati equation is scheduled according to the magnitude of a virtual control signal. The scheduling algorithm allows saturation up to a certain level and does not rely on confining the state inside ellipsoidal invariant sets. Computational requirements of the algorithm are not expensive. The control scheme guarantees £2 stability from a disturbance input to the state, and global asymptotic stability in the absence of disturbance. Moreover, the scheme is robust to a class of static input nonlinearities and an additional design is provided that guarantees robustness against a class of input-additive uncertainties.
7. REFERENCES Fuller, A.T. (1969). In-the-Iarge stability of relay and saturating control systems with linear controller. Int. Jornal of Control 10, 457-480.
Lauvdal, T . (1998). Stabilization of Linear Systems with Input Magnitude and Rate Saturations. PhD thesis. Norwegian University of Science and Technology. Norway. Lin, Z. (1998). Global control of linear systems with saturating actuators. A utomatica 34(7), 897-905. Lin, Z. and A. Saberi (1993). Semi-global exponential stabilization of linear sytems subject to input saturation via linear feed backs. Syst . Cont. Letters 21(3), 225-239. Megretski, A. (1996). £2 BIBO output feedback stabilization with saturated control. In: 13th Triennial IFAC World Congress. San Francisco, USA. pp. 435-440. Reginatto, R., A.R. Teel and E.R. De Pieri (2000). Stabilization of saturated linear systems: A scheduling mechanism based on the control signal. Technical report . Federal University of Santa Catarina. Florianopolis, Brazil. Saberi, A., Z. Lin and A.R. Teel (1996). Control of linear systems with saturating actuators . IEEE Trans. Aut. Cont. 41(3), 368-378. Sontag, E.D. and H.J . Sussmann (1990). Nonlinear output feedback design for linear systems with saturating actuators. In: 29th CDC. Honolulu, Hawaii. pp. 3414-3416. Suares, R. et all (1997) . £2 disturbance attenuation for linear systems with bounded controls: an ARE-based approach. In: Control of Uncertains Systems with Bounded Controls (S. Taubouriech & G . Garcia, Ed.) . Springer. Sussmann, H.J. and Y. Yang (1991). On the stabilization of multiple integrators by means of bounded feedback controls. In: 30th CDC. Brighton, England. pp. 70-72. Sussmann, H.J., E.D. Sontag and Y. Yang (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Trans . Aut. Cont. 39(12),2411-2424. Teel, A.R. (1992). Global stabilization and restricted tracking for multiple integrators with bounded controls. Syst. Cont. Letters 18, 165-17l. Teel, A.R. (1995a). Linear systems with input nonlinearities: Global stabilization by scheduling a family of ll oo -type controllers. Int. J . Robust Nonlinear Control 5, 399-41l. Teel, A.R. (1995b). Semi-global stabilizability for linear null controllable systems with input nonlinearities. IEEE Trans . A ut. Cont. 40(1),96-100. Teel, A.R. (1996). A nonlinear small gain theorem for the analysis of control systems with saturation. IEEE Trans. Aut. Cont. 41(9), 12561270. Wredenhagen, G.F. and P.R. Belanger (1994). Piecewise-linear LQ control for systems with input constraints. A utomatica 30(3), 403416.
432