On-off Control: Audio Applications Graham C. Goodwin Day 4: Lecture 3 16th September 2004 International Summer School Grenoble, France Centre for Complex Dynamic Systems and Control
1 Background In this lecture we address the issue of control when the decision variables must satisfy a finite set constraint. Finite alphabet control occurs in many practical situations including: on-off control, relay control, control where quantisation effects are important (in principle this covers all digital control systems and control systems over digital communication networks), and switching control of the type found in power electronics. Centre for Complex Dynamic Systems and Control
Exactly the same design methodologies can be applied in other areas; for example, the following problems can be directly formulated as finite alphabet control problems: quantisation of audio signals for compact disc production; design of filters where the coefficients are restricted to belong to a finite set (it is common in digital signal processing to use coefficients that are powers of two to facilitate implementation issues); design of digital-to-analog [D/A] and analog-to-digital [A/D] converters. Centre for Complex Dynamic Systems and Control
2. Finite Alphabet Control Consider a linear system having a scalar input u k and state vector x k ∈ R n described by x k + 1 = Ax k + Bu k . (1) A key consideration here is that the input is restricted to belong to the finite set U = { s 1 , s 2 , . . . , s n U } , (2) where s i ∈ R and s i < s i + 1 for i = 1 , 2 , . . . , n U − 1. Centre for Complex Dynamic Systems and Control
We will formulate the input design problem as a receding horizon quadratic regulator problem with finite set constraints. Thus, given the state x k = x , we seek the optimising sequence of present and future control inputs: u ( x ) � arg min u k ∈ U N V N ( x , u k ) , (3) where u k u k + 1 U N � U × · · · × U . u k � , (4) . . . u k + N − 1 Centre for Complex Dynamic Systems and Control
V N is the finite horizon quadratic objective function k + N − 1 V N ( x , u k ) � � x k + N � 2 � ( � x t � 2 Q + � u t � 2 P + R ) , (5) t = k with Q = Q > 0, P = P > 0, R = R > 0 and where x k = x . Centre for Complex Dynamic Systems and Control
Following the usual receding horizon principle, only the first control action, namely � � u ( x ) , u ( x ) � 1 0 · · · 0 (6) is applied. At the next time instant, the optimisation is repeated with a new initial state and the finite horizon window shifted by one. Centre for Complex Dynamic Systems and Control
3. Nearest Neighbour Characterisation of the Solution Since the constraint set U N is finite, the optimisation problem (3) is nonconvex. Indeed, it is a hard combinatorial optimisation problem whose solution requires a computation time that is exponential in the horizon length. Thus, one needs either to use a relatively small horizon or to resort to approximate solutions. We will adopt the former strategy based on the premise that, due to the receding horizon technique, the first decision variable is all that is of interest. Moreover, it is a practical observation that this first decision variable is often insensitive to increasing the horizon length beyond some relative modest value. Centre for Complex Dynamic Systems and Control
We vectorise the objective function as follows: Define x k + 1 B 0 . . . 0 0 A A 2 x k + 2 AB B . . . 0 0 x k � Φ � Λ � , , , . . . . . . ... . . . . . . . . . . . . A N − 1 B A N − 2 B A N x k + N AB B . . . (7) Centre for Complex Dynamic Systems and Control
Given x k = x the predictor x k satisfies x k = Φ u k + Λ x . (8) Hence, the objective function can be re-written as V N ( x , u k ) = ¯ V N ( x ) + u k H u k + 2 u k Fx , (9) where H � Φ Q Φ + R ∈ R N × N , F � Φ Q Λ ∈ R N × n , Q � diag { Q , . . . , Q , P } ∈ R Nn × Nn , R � diag { R , . . . , R } ∈ R N × N , V N ( x ) does not depend upon u k . and ¯ Centre for Complex Dynamic Systems and Control
By direct calculation, it follows that the minimiser, without taking into account any constraints on u k , is u ( x ) = − H − 1 Fx . (10) Centre for Complex Dynamic Systems and Control
Definition: Nearest Neighbour Vector Quantiser Given a countable set of nonequal vectors B = { b 1 , b 2 , . . . } ⊂ R n B , the nearest neighbour quantiser is defined as a mapping q B : R n B → B that assigns to each vector c ∈ R n B the closest element of B (as measured by the Euclidean norm), that is, q B ( c ) = b i ∈ B if and only if c belongs to the region c ∈ R n B : � c − b i � 2 ≤ � c − b j � 2 for all b j � b i , b j ∈ B � � c ∈ R n B : there exists j < i such that � c − b i � 2 = � c − b j � 2 � � \ . (11) Centre for Complex Dynamic Systems and Control
In order to simplify the problem, we introduce the same coordinate transformation utilised earlier, that is, the one that turns the cost contours into (hper) spheres. u k = H 1 / 2 u k , ˜ (12) which transforms the constraint set U N into ˜ N . U Centre for Complex Dynamic Systems and Control
The optimiser u ( x ) can be defined in terms of this auxiliary variable as u ( x ) = H − 1 / 2 arg min u k ) , U N J N ( x , ˜ (13) u k ∈ ˜ ˜ where u k ) � ˜ u u k + 2 ˜ u k H − / 2 Fx . J N ( x , ˜ k ˜ (14) Centre for Complex Dynamic Systems and Control
The level sets of J N are spheres in R N , centred at u ( x ) � − H − / 2 Fx . ˜ (15) Centre for Complex Dynamic Systems and Control
Hence, the constrained optimiser (3) is given by the nearest u neighbour to ˜ ( x ) , namely u k ) = q ˜ U N ( − H − / 2 Fx ) . U N J N ( x , ˜ arg min (16) u k ∈ ˜ ˜ Centre for Complex Dynamic Systems and Control
Summary Theorem: Closed Form Solution Let U N = { v 1 , v 2 , . . . , v r } , where r = ( n U ) N . Then the optimiser u ( x ) in (3) is given by u ( x ) = H − 1 / 2 q ˜ U N ( − H − / 2 Fx ) , (17) U N ( · ) maps R N to ˜ N , where the nearest neighbour quantiser q ˜ U defined as N � { ˜ v i = H 1 / 2 v i , v i ∈ U N . ˜ v 1 , ˜ v 2 , . . . , ˜ v r } , ˜ (18) U Centre for Complex Dynamic Systems and Control
The receding horizon controller satisfies � � H − 1 / 2 q ˜ U N ( − H − / 2 Fx ) . u ( x ) = 1 0 · · · 0 (19) This solution can be illustrated as the composition of the following transformations: ∈ R N H − 1 2 q ˜ U N ( · ) x ∈ R n − H − 2 F → u ∈ U N [ 1 0 · · · 0 ] → u ∈ U . u − − − − − − − → ˜ − − − − − − − − − − − − − − − − − − − − (20) Centre for Complex Dynamic Systems and Control
4. State Space Partition The optimal expression partitions the domain of the quantiser into polyhedra, called a Voronoi partition . Since the constrained optimiser u ( x ) is defined in terms of q ˜ U N ( · ) , an equivalent partition of the state space can be derived. Centre for Complex Dynamic Systems and Control
Theorem The constrained optimising sequence u ( x ) can be characterised as u ( x ) = v i ⇐⇒ x ∈ R i , where z ∈ R n : 2 ( v i − v j ) Fz ≤ � v j � 2 � H for all v j � v i , v j ∈ U N � H − � v i � 2 R i � z ∈ R n : there exists j < i such that 2 ( v i − v j ) Fz = � v j � 2 � � H − � v i � 2 \ . H (21) Centre for Complex Dynamic Systems and Control
5. Examples: 5.1 Open Loop Stable Plant Consider an open loop stable plant described by � � � � 0 . 1 2 0 . 1 x k + 1 = x k + u k , (22) 0 0 . 8 0 . 1 and the binary constraint set U = {− 1 , 1 } . The receding horizon control law with R = 0 and � � 1 0 P = Q = , (23) 0 1 partitions the state space into the regions depicted in the next figure, for constraint horizons N = 2 and N = 3. Centre for Complex Dynamic Systems and Control
N = 2 R 0.5 4 R 2 x 2 0 k R 3 R −0.5 1 −80 −60 −40 −20 0 20 40 60 80 N = 3 R 8 0.5 R 4 R 7 x 2 R 0 R 6 k R 3 2 R 5 −0.5 R 1 −80 −60 −40 −20 0 20 40 60 80 x 1 k Figure: State space partition for the plant (22). Centre for Complex Dynamic Systems and Control
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