Introduction to Constrained Control Graham C. Goodwin September 2004 Centre for Complex Dynamic Systems and Control
1.1 Background Most of the literature on Control Theory deals with Linear Unconstrained Systems . However to get the most out of a system, we usually need to deal with nonlinearities . The most common nonlinearity met in practice are Actuator Limits . Centre for Complex Dynamic Systems and Control
To get the most out of a system you need to push up against limits. Centre for Complex Dynamic Systems and Control
Other examples? Playing sport at international level Excelling in business or academia Aerospace, chemical process control, . . . Control is a key enabling technology in many (all?) areas Getting the most out of control means pushing against boundaries Centre for Complex Dynamic Systems and Control
1.2 Approaches to Constrained Control Cautious (back off performance demands so constraints are not met) Serendipitous (allow occasional constraint violation) Evolutionary (begin with a linear design and add embellishments, for example, antiwindup) Tactical (include constraints from the beginning, for example, MPC) Centre for Complex Dynamic Systems and Control
1.3 Example: Rudder Roll Stabilisation of Ships (See lecture 3.5) It has been observed that unless appropriate actions are taken to deal with constraints, then the performance of rudder roll stabilisation systems can be worse than if nothing is done due to the effect of actuator amplitude and slew rate constraints. Centre for Complex Dynamic Systems and Control
1.4 Model Predictive Control Model Predictive Control (MPC) is a prime example of tactical method. Long history in petrochemical industry. Many thousands of applications. Several commercial products. Industrial “credibility”. Centre for Complex Dynamic Systems and Control
Background A survey by Mayne et al. (2000) divides the literature on MPC in three categories: Theoretical foundations: the optimal control literature. Dynamic Programming (Bellman 1957), the Maximum Principle (for example, Lee & Markus 1967). “Process control” literature, responsible for MPC’s adoption by industry. Evolving generations of MPC technology. An example of practice leading theory. “Modern” literature, dealing with theoretical advances such as stability and robustness. Centre for Complex Dynamic Systems and Control
General Description MPC is a control strategy which, for a model of the system, optimises performance (measured through a cost function ) subject to constraints on the inputs, outputs and/or internal states. Due to the presence of constraints it is difficult, in general, to obtain closed formulae that solve the above control problem. Hence, MPC has traditionally solved the optimisation problem on line over a finite horizon using the receding horizon technique. This has also restricted the applicability of MPC to processes with slow time constants that allow the optimisation to be solved on line. Recent results allow faster systems to be handled. Centre for Complex Dynamic Systems and Control
An Illustrative Example We will base our design on linear quadratic regulator [LQR] theory. Thus, consider an objective function of the form: N − 1 V N ( { x k } , { u k } ) � 1 N Px N + 1 � � � 2 x x k Qx k + u k Ru k (1) , 2 k = 0 where { u k } denotes the control sequence { u 0 , u 1 , . . . , u N − 1 } , and { x k } denotes the corresponding state sequence { x 0 , x 1 , . . . , x N } . In (1), { u k } and { x k } are related by the linear state equation: x k + 1 = Ax k + Bu k , k = 0 , 1 , . . . , N − 1 , where x 0 , the initial state, is assumed to be known. Centre for Complex Dynamic Systems and Control
The following parameters allow one to influence performance: the optimisation horizon N the state weighting matrix Q the control weighting matrix R the terminal state weighting matrix P For example, reducing R gives less weight on control effort, hence faster response. R → 0 is called “cheap control”. Centre for Complex Dynamic Systems and Control
Details of Example Consider the specific linear system: x k + 1 = Ax k + Bu k , (2) y k = Cx k , with � � � � 1 1 0 . 5 � � 1 0 A = B = C = , , , 0 1 1 which is the zero-order hold discretisation with sampling period 1 of the double integrator d 2 y ( t ) = u ( t ) . dt 2 Centre for Complex Dynamic Systems and Control
Example u k x k linear controller sat system Figure: Feedback control loop for Example 1 if u > 1 , sat( u ) � u if | u | ≤ 1 , (3) − 1 if u < − 1 . Centre for Complex Dynamic Systems and Control
(i) Cautious Design � � 1 0 ( N = ∞ , P = 0) and weighting matrices Q = C C = and 0 0 R = 20 gives the linear state feedback law: � � u k = − Kx k = − 0 . 1603 0 . 5662 x k . Centre for Complex Dynamic Systems and Control
Cautious Design 1 0.8 0.6 u k 0.4 0.2 0 −0.2 −0.4 0 5 10 15 20 25 k 1 0 −1 −2 y k −3 −4 −5 −6 0 5 10 15 20 25 k Figure: u k and y k for the cautious design u k = − Kx k with weights Q = C C and R = 20. Centre for Complex Dynamic Systems and Control
(ii) Serendipitous Design Using the same Q = C C in the infinite horizon objective function we try to obtain a faster response by reducing the control weight to R = 2. We expect that this will lead to a control law having “higher gain.” Centre for Complex Dynamic Systems and Control
Serendipitous Design 3 2 u k 1 0 −1 0 5 10 15 20 25 k 1 0 −1 y k −2 −3 −4 −5 −6 0 5 10 15 20 25 k Figure: u k and y k for the unconstrained LQR design u k = − Kx k (dashed line), and for the serendipitous strategy u k = − sat( Kx k ) (circle-solid line), with weights Q = C C and R = 2. Centre for Complex Dynamic Systems and Control
Encouraged by the above result, we might be tempted to “push our luck” even further and aim for an even faster response by further reducing the weighting on the input signal. Accordingly, we decrease the control weighting in the LQR design even further, for example, to R = 0 . 1. Centre for Complex Dynamic Systems and Control
6 4 2 u k 0 −2 −4 −6 0 5 10 15 20 25 k 4 2 0 y k −2 −4 −6 0 5 10 15 20 25 k Figure: u k and y k for the unconstrained LQR design u k = − Kx k (dashed line), and for the serendipitous strategy u k = − sat( Kx k ) (circle-solid line), with weights Q = C C and R = 0 . 1. Centre for Complex Dynamic Systems and Control
The control law u = − sat( Kx ) partitions the state space into three regions in accordance with the definition of the saturation function (3). Hence, the serendipitous strategy can be characterised as a switched control strategy in the following way: − Kx if x ∈ R 0 , u = K ( x ) = 1 if x ∈ R 1 , (4) − 1 if x ∈ R 2 . Notice that this is simply an alternative way of describing the serendipitous strategy since for x ∈ R 0 the input actually lies between the saturation limits. The partition is shown in following figure. Centre for Complex Dynamic Systems and Control
Figure 5 4 3 2 1 x 2 k 0 −1 R 2 −2 R 0 −3 R 1 −4 x 1 −6 −4 −2 0 2 4 6 k Figure: State space trajectory and space partition for the serendipitous strategy u k = − sat( Kx k ) , with weights Q = C C and R = 0 . 1. Centre for Complex Dynamic Systems and Control
Examination of figure 8 suggests a heuristic argument as to why the serendipitous control law may not be performing well in this case. We can think, in this example, of x 2 as “velocity” and x 1 as “position.” Now, in our attempt to change the position rapidly (from − 6 to 0), the velocity has been allowed to grow to a relatively high level ( + 3). This would be fine if the braking action were unconstrained. However, our input (including braking) is limited to the range [ − 1 , 1 ] . Hence, the available braking is inadequate to “pull the system up”, and overshoot occurs. Centre for Complex Dynamic Systems and Control
(iii) Tactical Design Perhaps the above heuristic argument gives us some insight into how we could remedy the problem. A sensible idea would seem to be to try to “look ahead” and take account of future input constraints (that is, the limited braking authority available). To test this idea, we take the objective function (1) as a starting point. Centre for Complex Dynamic Systems and Control
Tactical Design We use a prediction horizon N = 2 and minimise, at each sampling instant i and for the current state x i , the two-step objective function: i + 1 V 2 ( { x k } , { u k } ) = 1 i + 2 Px i + 2 + 1 � � � 2 x x k Qx k + u k Ru k (5) , 2 k = i subject to the equality and inequality constraints: x k + 1 = Ax k + Bu k , (6) | u k | ≤ 1 , for k = i and k = i + 1. Centre for Complex Dynamic Systems and Control
Tactical Design In the objective function (5), we set, as before, Q = C C , R = 0 . 1. The terminal state weighting matrix P is taken to be the solution of the Riccati equation P = A PA + Q − K ( R + B PB ) K , where K = ( R + B PB ) − 1 B PA is the corresponding gain. Centre for Complex Dynamic Systems and Control
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